Methods to Calculate Stresses Resultingfrom Whiplash Injuries

1.                  Introduction

There is a significant divergence of opinions on the probability and potential for injury to the spinal column resulting from rear-end vehicular collisions. Some experts claim that injuries may occur as a result of any speed change. We do not agree with such a position because, the principles of material mechanics are well understood and utilized in design and failure analysis irrespective of the material being studied. Therefore, this treatise affords the forensic biomechanical engineer a basis for calculating the potential for failure and injury based on known scientific data.

Generally, these spinal injuries are referred to as whiplash type maladies that are often claimed to result to the cervical, thoracic, and lumbar sections of the spine. It should be noted that the modern vehicle has substantial padding that affects the lumbar and thoracic spine. Consequently, significant speed changes are required to overcome the supportive cushioning effect of the seat back. Furthermore, the seats, especially for the front passengers, are designed to give or collapse beyond a certain limit. Please note that in this analysis we are not concerned with the crashworthiness of the vehicles, seats, and restraint systems.

Similarly, vehicles are equipped with head rests that also offer some degree of protection when the head and the neck of the occupant is thrust rear-ward in the collision. Depending on the seat configuration and the physical structure of the occupant, the head rest may offer little or no protection. As a further caveat for this analysis we note that the supportive structure of the head rest is simply not considered.

(Figure 1) represents a spinal column superimposed on a typical seat. Two important features are noted. One, the spinal column is much more robust in the lower thoracic and lumbar regions. Two, the maximum exposure to whiplash can occur to the cervical and portions of the upper thoracic regions.

1.1.  Data on Injury to Soft Tissue Spinal Structures

(Tables 1&2) show spinal column dimensions that are necessary for most biomechanical whiplash calculations. From a mechanical standpoint, it is well understood that larger structures, whether biological or not, are more resilient and require greater forces to overcome their failure limits.

The data above was taken from H Yamada, and H Franck and D Franck. (Figure 2) shows the stress-strain curves for wet intervertebral discs in tension for various portions of the spine for individuals between the ages of 20 and 39 years of age corresponding to the data on (Table 3). Note that the cervical discs exhibit the greatest stress capability simply because they must allow for the greatest movement in the neck and head area. I fact the cervical discs allow up to a 40-degree angle of twist while the lumbar and thoracic vertebrae restrict the angle of twist to about 20 degrees.

2.      Hooke’s Law

(Figure 2)requires some explanation and review. Analysis of structural elements, in this case the stresses on the spinal disc structures, requires a determination of the forces acting on the structure. The forces that affect the structure and the resistance of the particular structure to separation of movement within the structure are considered internal. The forces that transmit the loads are considered external and are dependent on the motion produced on the entire body by an event such as a rear-end collision. Generally, the force acting on the body is proportional to the stress times the cross-sectional area or

The Second Method is that of a couple which is produced by the movement of the head resulting from the collision. This is a two-dimensional problem. Consider Figure 3 below.

In the Couple Model  is the force produced by the collision which produces a moment M on the adjoining vertebra and damages the disc. Standard equations are,

The cross section of the disc for this model is not the average of all the discs but the value for the average of the cervical and thoracic discs which is approximately 350 .

The Third Method is produced by a torque on the discs. One argument that is proposed states that the rotation of the head relative to the direction of the force makes a difference whether injury might occur. (Figure 4) represents the torque produced by such motion. The pertinent equations follow,

For the torque equations t is the pulse width but the 2r term is different as follows. The separation between the vertebrae at mid line is approximately 20 mm and the area of the cervical discs is about 350 mm² with a diameter of approximately 18 mm. Consequently, the 2r term is about 0.176 ft.

3.1.  Summary of Whiplash Tests

3.2.  The Graphs for the Previous Tests are Shown Below

3.3.  The Graphs for the Tests on Table 7 are Shown below

For each of the test graphs, the top graph in red corresponds to the head acceleration and the bottom graph represents the measurement of the chest acceleration.

The data from Table 8 is plotted below.The Couple Model yields the greatest stress shown in yellow. The Damask and Torque Models yield essentially the same results.

4.      Conclusions

In order to assess the potential for injury to the upper spinal column, calculations are required in scientifically correct terms

Figure 1: Spinal Column and Seat.

Figure 2: Elongation Tensile Properties of Discs.

Figure 3: Couple Model.

Figure 4:Torque Model.

Figure 5: Stress Comparison.

Table 1: Height of Human Vertebraeby Age and Section (mm).

Table 2: Cross Sectional Area of Human Vertebrae by Age and Section (mm²).

(Tables 3,4&5) relate the ultimate strengths in tension, torsion, and compression for various sections of the spine and for various age groups.

Table 3: Ultimate Tensile Strength of Human Intervertebral Discs (Kg/mm²).

Table 4:Ultimate Torsional Strength of Human Intervertebral Discs (Kg/mm²).

Table 5: Ultimate Compressive Strength of Human Intervertebral Discs (Kg/mm²).

Table 6: Whiplash Tests and Head Accelerations 40-year-old Female.

Table 7: Whiplash Tests and Head Accelerations 66-year-old Male.

Table 8:Comparison of Methods.

1.       Damask AC, Damask J (1990) InjuryCausation Analysis: Case Studies and Data Sources.The Michie Company,Charlottesville, VA.

2.       H. Franck, D Franck(2016) Forensic Biomechanics and Human Injury – Criminal and Civil Applications-An Engineering Approach. Boca Raton FL: 262.

3.       Yamada H (1970)The Strength of Biological Materials. Williams and Wilkins,Philadelphia, PA.