Here is a link to a PDF of this article: Rose-Fracture-Energy-Calculations

**Introduction**

The following mechanisms of energy dissipation may need to be considered when calculating the initial speed of a vehicle that impacted a wooden utility pole: (1) crushing of the vehicle; (2) full or partial fracture of the pole; (3) moving and tilting of the pole within the ground; (4) acceleration of the pole after a full fracture; and (5) tire, and other dragging forces, acting on the vehicle during its post-impact motion [Daily, 2009; Cofone, 2007 and 2012]. Each of these will be discussed in this article. However, the focus will be on validating Kent and Strothers’ method for calculating the full or partial fracture energy for the pole [1998].

In general, a vehicle’s kinetic energy at the time of impact with a wooden utility pole can be calculated by adding the vehicle’s post impact kinetic energy to the energies dissipated through vehicle crush, pole fracture, and pole movement (including tilting and accelerating after fracture). This is described mathematically by Equation (1).

In Equation (1), the variables have the following meaning:

Once the kinetic energy at impact with the pole is obtained, this kinetic energy can be converted to the impact speed with the following equation.

In Equation (2), the variables have the following meaning:

Crushing of the Vehicle – An extensive literature covers methods for calculating the energy dissipated during crushing of the vehicle. For the purpose of this study, we relied on the crash tests cited and analyzed by Craig in *Accident Reconstruction Journal*[1993, 1995a, 1995b, 1995c, 1995d]. These tests were passenger car impacts with rigid pole barriers, thus they were tests in which no energy was dissipated by moving, deforming, or fracturing the pole. All of the collision energy could be reasonably assumed to have been dissipated through crushing of the vehicle. Craig had analyzed this data by plotting the maximum vehicle crush versus the equivalent impact speed for each test. Some of the tests were repeated impacts of the same vehicle into the pole barrier, so for these tests, the equivalent impact speed was calculated by adding up the collision energies from the impacts to which the vehicle had been subjected.

Rather than rely on Craig’s previous analysis of this data, we conducted our own curve-fitting analysis with Microsoft Excel. The graph of Figure 1 shows the data from the Craig publications with the maximum vehicle crush plotted on the horizontal axis in inches and the impact velocity plotted on the vertical axis in miles per hour. This graph also depicts the second-order polynomial that we fit to the data, which had the following equation:

In Equation (3), the variables have the following meaning:

When applied to the analysis of a real-world crash, where there are energy dissipation mechanisms other than the vehicle crush, the speed calculated with Equation (3) will be the Equivalent Barrier Speed (EBS), not the impact speed.

The coefficient of determination for Equation (3) was 0.9417. To quantify the uncertainty in speeds calculated with this equation, an upper and lower bound was established that would encompass all of the data plotted in Figure 1. These curves are depicted with dotted lines in Figure 1. Through this process, we concluded that all of the data was captured by including a ±5.5 mph uncertainty on speeds calculated with Equation (3). Clearly, there cannot be 5 inches of crush for an impact speed of 0 miles per hour, so this uncertainty would narrow at low impact speeds. Overall, though, this range of uncertainty is not surprising considering the fact that this approach of lumping all of these vehicles into one dataset ignores the structural differences that would clearly exist between these vehicles [Varat and Husher, 1999].

Moving and Tilting of the Pole – Daily [2009] and Cofone [2007 and 2012] discuss a method for calculating the energy dissipated by moving and tilting the pole within the ground. This method is based on Newton’s Third Law, which requires that the force applied to the pole by the vehicle is equal in magnitude and opposite in direction to the force applied to the vehicle by the pole. If the collision force can be calculated based on the vehicle deformation, and then the distance through which that force acted can be determined (this would be the pole displacement at the level of the force application), then the energy dissipated in displacing the pole can be calculated. Daily reports that this energy can be calculated with the following equation:

In Equation (4), the variables have the following meaning:

The effective force coefficient is a multiplier that accounts for the changing orientation of the force applied to the pole as it rotates. Daily’s publication gives instructions for calculating this factor. Daily recommends using methods of crush analysis to calculate the average collision force for this calculation, though other methods could also be used.

Post-Impact Motion of the Vehicle – There are many references that describe methods for calculating the energy dissipated during the post-impact movement of a vehicle. Carter [2012], for instance, discussed several methods for calculating the energy dissipated while a vehicle yaws across the road surface. Rose [2016] discussed the effects of high post-impact spin rates within this analysis and also discussed the use of simulation for quantifying the post-impact energy dissipation.

Fracturing of the Pole – Several studies have discussed methods for calculating the energy dissipated by full or partial fracture of a utility pole. The discuss here will focus only on the 1998 study by Kent and Strother, in which they made “a first attempt to understand the energy absorbing processes operating when vehicles strike trees and wooden poles in order to make reasonable estimates of the magnitude of the tree/pole energy dissipated in the crash.” Since the time Kent and Strother published their study, additional utility pole impact tests have been added to the literature that can be used to validate their approach [Croteau, 2011].

Kent and Strother derived the following expression, which yields the energy to staticallyinitiate fracture of a wooden utility pole:

In this equation, the variables have the following meaning:

The modulus of rupture and the *longitudinal* modulus of elasticity for the relevant wood species can be obtained from the Wood Handbook, which is produced by the Forest Products Laboratory of the United States Department of Agriculture Forest Service. A PDF version of this handbook is accessible for free via a Google search. The Wood Handbook does not report the *radial* modulus of elasticity for the various wood species, but it does report the ratio of the radial to the longitudinal, so the radial modulus of rupture can be calculated. The cross-sectional moment of inertia of the pole can be calculated with the following equation.

In this equation:

Testing and modeling reported by Kent and Strother showed that the dynamic energy required to fracture a wooden utility pole is greater than the static fracture energy, so Equation (5) will underestimate the fracture energy for a vehicle impacting a utility pole. In their article, Kent and Strother reported the difference between the static and dynamic fracture energies for various pole diameters (see Table 8 in their article). To obtain a reasonable approximation of the degree to which the energy calculated by Equation (5) should be increased to represent dynamic loading of the pole, we fit a line to the Kent and Strother data. The result was the following equation:

In this equation:

The coefficient of determination for this equation was 0.7562.

In addition to the issue of static versus dynamic fracture initiation energy, Kent and Strother noted that “several confounding issues must be addressed when dealing with wood structures. Properties of wood not only vary greatly depending upon species, but they also can be strongly dependent on geographical location, moisture content, grain orientation, grain coarseness, texture, hard mineral deposits, grain irregularities, and treatment.” They state that, of these variables, moisture content is perhaps the most influential and they present the following equation that can be used to adjust any of the mechanical properties of the wood, based on the moisture content. This equation is also listed in the Wood Handbook.

In this equation:

So far, this section has laid out a method for calculating the energy required to initiatefracture of the pole. But what about the energy required to fully fracture the pole? Based on pendulum testing conducted by Kent and Strother with 1/8th scale poles, they concluded that “a second-order polynomial fit most accurately represents the relationship between the energy to completely fracture the pole specimens and the specimen moment of inertia…” They presented the following equation for calculating the fracture energy for poles constructed of southern yellow pine:

For poles that completely fracture, Kent and Strother recommend, first, calculating the energy required to initiate the fracture [Equations (5) through (8)]. If the pole is not made of southern yellow pine, then they recommend next calculating the fracture initiation energy for the same size pole as if it were made of southern yellow pine, then based on the ratio of the fracture initiation energies for the actual pole and the southern yellow pine pole, scaling the result of Equation (9) to obtain the total fracture initiation energy. The flow chart of Figure 2 summarizes the approach they propose.

**Validating the Kent and Strother Approach**

In 2011, Croteau reported seven full-scale crash tests that involved moving barriers impacting wooden utility poles. Five of these tests involved *non-deformable* moving barriers and two involved *deformable* moving barriers. The face of the deformable barrier had a stiffness that was representative of the frontal stiffness of a passenger car. Both barriers had a bumper height of 26 inches. The class 4 utility poles that were tested were constructed of southern yellow pine and were coated with Pentachlorophenol preservative. They were 35 feet tall with a nominal diameter of 10 inches and were buried six feet deep. They were mounted in hard-packed desert soil.

After the tests, the moisture content of each pole was measured. The authors also reported that the poles were surrounded by a four-inch thick reinforced concrete pad to replicate a typical sidewalk. This concrete prevented the poles from moving and tilting within the soil, and so, there was no need to consider these energy dissipation mechanisms in our analysis of these tests. The non-deformable barrier tests also eliminated the need to consider vehicle deformation, and since Croteau reported the vehicle speeds immediately following pole fracture, there was no need to calculate the post-impact speed. This allowed the pole fracture energy to be isolated in the test data and in the calculations.

Table 1 lists the barrier impact speeds and the pole moisture contents for the five tests that utilized a non-deformable barrier. This table also lists the actual change in velocity that the barrier experienced in each test (up to the point of pole fracture), the post-fracture barrier speed, and the actual energy dissipated in fracturing the pole. Table 2 lists the barrier impact speeds and the pole moisture contents for the two tests that utilized a deformable barrier. All five of the crash tests with the non-deformable barrier resulted in complete fracture of the pole. Of the two tests with a deformable moving barrier, one pole fractured (Test #4 @ 33.2 mph) and one did not (Test #6 @ 14.8 mph impact speed).

Calculations for the Tests with the Non-Deformable Barriers: We used Equations (5), (6), (7), and (8) to calculate the dynamic fracture initiation energy for the poles involved in the tests with the non-deformable barriers. There are four species of southern yellow pine (shortleaf, slash, longleaf, and loblolly). Croteau did not report of which species the poles in his tests were constructed. Thus, in obtaining material properties from the Wood Handbook, we obtained the low and high values amongst these four species and used them to establish a low-end and high-end calculation of the dynamic fracture initiation energy. Table 3 lists the range of values employed in the analysis for the modulus of rupture. The second and third columns list this modulus for green wood, the fourth and fifth for wood with a 12% moisture content, and the sixth and seventh for the actual reported moisture contents, with the adjustment being made using Equation (8). The adjusted values based on the actual moisture contents were the ones actually used in the calculations. Tables 4 lists similar values for the longitudinal modulus of elasticity and Table 5 for the radial modulus of elasticity.

After calculating a range for the fracture initiation energy, we used Equation (9) to calculate the total fracture energy, since all of 5 of the poles fractured completely. Since these poles were made of southern yellow pine, there was no need to adjust the energy calculated with Equation (9). Table 6 lists the actual DV and energy dissipation during the pole fracture for each of the non-deformable barrier tests along with the range of these values calculated using the Kent and Strother approach. For all five cases, the entire range of calculated fracture initiation energies was less than the total fracture energy, as would be expected. In all but one case, the total fracture energy calculated with Equation (9) overestimated the actual total fracture energy. However, all of the calculated barrier velocity changes were within 0.9 mph of the actual value.

Calculations for the Tests with the Deformable Barriers: Similar calculations were carried out for the cases involving deformable moving barriers. However, for these cases, in addition to using Equations (5), (6), (7), and (8) to calculate the dynamic fracture initiation energy for the poles, we also Equation (3) to calculate the equivalent barrier speed for the crushing of the barrier. For the test at 14.8 mph (Test #6), Equation (3) yielded an equivalent barrier speed of 13.1 mph, with a range of 7.6 to 18.6 mph. The mean of this range is 1.7 mph below the actual impact speed, but the range encompasses the actual value. The pole did not fracture in this case and this would likely be an instance in practice where a reconstructionist would neglect the energy that went into deforming the fibers of the pole. In describing the damage to this pole after the test, though, Croteau noted: “Creasing of fibers is apparent in the bumper contact area and outer-layer fibers on the non-struck side began to pull apart in one small area.” Realistically, there was some energy absorbed by this pole, but it was below the level necessary to initiate fracture. Application of the Kent and Strother fracture initiationmodel to estimate this energy resulted in a speed range for this case of 14.6 to 16.3 mph, a range that contains the actual speed (this range neglects the uncertainty in the calculation of the equivalent barrier speed for the barrier).

For the test at 33.2 mph (Test #4), Equation (3) yielded an equivalent barrier speed of 16.0 mph (10.5 mph to 21.5 mph). Following the fracture of the pole, the deformable barrier was still moving at 27.3 mph. Application of the Kent and Strother model to estimate the energy dissipated in fracturing the pole and combining that energy with the crush and post-impact energies resulted in a calculated speed range for this case of 32.1 to 32.6 mph, 0.6 to 1.1 mph below the actual speed. This range neglects the uncertainty in the calculation of the equivalent barrier speed for the barrier deformation. If this was included, the full range would encompass the actual impact speed. For this case, Croteau reported a total energy dissipation from the barrier crush and pole fracture of 52,327 foot-pounds. Calculations with Equation (3) and with the Kent and Strother approach to calculating the dynamic pole fracture initiation energy yielded a total energy dissipation between 42,864 and 47,600 foot-pounds.

**Discussion and Conclusions**

When reconstructing a vehicle collision with a wooden utility pole, the following mechanisms of energy dissipation may need to be considered: (1) crushing of the vehicle; (2) full or partial fracture of the pole; (3) moving and tilting of the pole within the ground; (4) acceleration of the pole after a full fracture; and (5) tire, and other dragging forces, acting on the vehicle during its post-impact motion. One assessment that the reconstructionist will have to make for any particular case is the significance of the energy absorbed by the pole. In instances where the pole does not fracture (even partially), this energy can reasonably be neglected. However, in instances when the pole fully or partially fractures, the method proposed by Kent and Strother provides a method for calculating the likely energy absorbed by the pole.

This study has examined the degree to which the Kent and Strother method can reasonably be relied on to accurately predicted pole fracture energies. To make this assessment, 7 full-scale tests reported by Croteau were utilized. For all 6 tests in which the utility poles fractured, the Kent and Strothers model predicted a fracture initiationenergy that was less than the actual pole fracture energy. This is as expected, since the total energy to fully fracture the pole would be expected to be greater than the energy to initiate the fracture. In 4 of the 6 tests in which the utility poles fully fractured, the Kent and Strother method overestimated the total fracture energy. For 2 of the tests with complete fracture, the model underestimated the total fracture energy. For the tests with the non-deformable barrier, all of the calculated barrier velocity changes were within 0.9 mph of the actual value. For the case with the deformable barrier where the pole did fracture, application of the Kent and Strother model to estimate the energy dissipated in fracturing the pole and combining that energy with the crush and post-impact energies resulted in a calculated speed range for this case of 32.1 to 32.6 mph, 0.6 to 1.1 mph below the actual speed (this neglects the uncertainty in the calculation of the equivalent barrier speed for the barrier deformation). For this case, calculations with Equation (3) and with the Kent and Strother approach to calculating the dynamic pole fracture initiation energy yielded an underestimate of the total energy dissipation.

For the case with the deformable barrier where the pole did not fracture, Equation (3) yielded an equivalent barrier speed of 13.1 mph, 1.7 mph below the actual impact speed. Because this would likely be an instance in practice where a reconstructionist would neglect the energy that went into deforming the fibers of the pole, reconstructionists would be likely to slightly underestimate the impact speed (this again neglects the uncertainty in the calculation of the barrier impact speed for the barrier crush). Croteau did note some damage to the fibers of the pole after this test. In this case, if a reconstructionist chose to try to incorporate some energy absorption by the pole through application of the Kent and Strother fracture initiation model, the result would be a speed range of 14.6 to 16.3 mph, a range that contains the actual speed, but generally overestimates the actual impact speed. Overall, the Kent and Strother method yielded dissipated energies and DVs that were reasonable, particularly when compared to the possibility of neglecting the energy absorbed by a fully or partially fractured pole, a practice that is unfortunately not uncommon.

**References**

1. Carter, Neal, Gray Beauchamp, Nathan A. Rose, “Comparison of Calculated Speeds for a Yawing and Braking Vehicle to Full-Scale Vehicle Tests,” Paper Number 2012-01-0620, Society of Automotive Engineers, 2012.

2. Cofone, Joseph N., The Investigation of Automobile Collisions with Wooden Utility Poles and Trees, 2nd Edition, Institute of Police Technology and Management, 2012.

3. Cofone, J., Rich A., Scott, J., “A Comparison of Equations for Estimating Speed Based on Maximum Static Deformation for Frontal Narrow Object Impacts,” Accident Reconstruction Journal, Volume 17, No. 6, December 2007.

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10. Daily, J.G., Daily, J.S., Rich, A.S., “A Method for Vehicle-Wooden Utility Pole Impact Speed Reconstruction,” Accident Reconstruction Journal, Vol. 19, No. 5, Sept/October, 2009.

11. Kent, Richard W., Charles E. Strother, “Wooden Pole Fracture Energy in Vehicle Impacts,” SAE Technical Paper 980214, 1998, doi:10.4271/980214.

12. Rose, Nathan A., Neal Carter, Gray Beauchamp, “Post-Impact Dynamics for Vehicles with a High Yaw Velocity,” SAE Technical Paper Number 2016-01-1470, 2016, doi:10.4271/2016-10-1470.

13. Varat, Michael S., Husher, Stein E., “Vehicle Crash Severity Assessment in Lateral Pole Impacts,” SAE Technical Paper Number 1999-01-0100, 1999, doi:10.4271/1999-01-0100.