Journal of Undergraduate Research
Volume 7, Issue 1 - September/October 2005

Examining the Variation in Modulus of Elasticity of Live Oak (Quercus virginiana Highrise™) Tree Trunks

Alison Trachet

INTRODUCTION

A tree trunk will deflect due to a bending moment caused by two dominant mechanical forces: gravitational load and wind load (Niklas 1996). The magnitude of this deflection is governed by the tree’s rigidity or modulus of elasticity (MOE), which is dependent upon the composition of the wood (Cannell and Morgan 1987). The reference value for the MOE of green live oak wood is based on a small, clear sample used in a simply supported beam test (Green et al. 1999). However, wood is a composite material that is anisotropic, and, therefore, the elasticity of wood might be better described by the structural modulus of elasticity, or SMOE (Spatz and Bruechert 2000).

The purpose of this experiment is to determine if there are statistical differences in the elastic properties of three-year-old live oak (Quercus virginiana Highrise™) trees 1) up the height of the trunk, 2) between the MOE and SMOE, 3) with varying moisture contents, and 4) with the bark still attached to the trunk. The MOE is not expected to differ significantly along the length of the trunk in young live oak trees, but the MOE of a small interior sample will likely be statistically different than that of a whole section of the trunk. Dry wood typically has a larger MOE than green wood. In addition, bark is a softer material than the interior wood and removing it should increase the SMOE.

METHODS AND MATERIALS

To confirm these hypotheses, two types of bend tests were performed: a three-point bend test and a cantilever bend test. Both tests were employed to determine the influence of moisture content on the elasticity of the wood. The trees tested were randomly selected from a group of forty genetically similar live oak trees. The trees were all of the same age, approximately the same size, and exposed to the same cultural practices during their growth. After harvest, dry wood samples were oven-dried at 70° C to a constant weight while green wood samples were stored in an ice bath and tested within twelve hours of harvest.

Coupons

The three-point bend test was comprised of a simply supported beam subjected to a concentrated load on the middle of the beam. The purpose of this test was to determine the MOE along the trunk as well as the MOE of different moisture contents. A 16”-long coupon with a 1” x 1” cross-section was formed from the middle of the tree trunk, in accordance with ASTM D 143-94 (2000). Two clear coupons were cut from each tree; one from the lower- and another from the upper-half of the trunk. Eight coupons were tested.

The coupons were placed on knife-edge supports fourteen inches apart in a Tinius-Olsen (TO) Super L 60 kip Universal Testing Machine (Figure 1). To prevent the supports from digging into the sample as the load was applied, a grooved section of aluminum channel rested on each support. This was replicated at the center to help distribute the point load. The concentrated load was applied to the center of the coupon at a rate of approximately two pounds/second until the sample displayed signs of failing. The vertical deflection of the coupon was measured at two, five, seven, nine, and twelve inches from the left support by five four-inch long Linear Variable Differential Transformers (LVDT) (604 R4K, BEI Technologies – Duncan Electronics Division, 15771 Red Hill Avenue, Tustin, CA 92780, USA). The deflection and load placed on the coupon were recorded by a National PCI 6031E data acquisition card (National Instruments, 11500 N Mopac Expwy, Austin, TX 78759, USA), and LabView 7.0 software (National Instruments).

Figure 1. Wood coupon subjected to compressive force in a Tinius-Olsen (TO) machine

Trunks

The purpose of the cantilever test was to determine the SMOE for whole-sections of the trunk (with and without bark) and to compare with those values with MOE values found in the coupon tests. The test was based on ASTM D 1036-99 “Standard Test Methods of Static Tests of Wood Poles.” However, this standard is designed for minimum twenty-foot long wood poles, so the dimensions of the test were scaled down to accommodate a four-foot long tree trunk (Figure 2). To account for the taper of the trunk, the diameter of the trunk was measured every six inches. Four oven-dried tree trunks were tested: two with the bark intact and two with the bark removed. The experiment was then repeated with four green tree trunks.

Figure 2. Fixed support of cantilever bend test

A worm gear actuator (M2002-1274, Duff-Norton Co, 9415 Pioneer Avenue, Charlotte, NC 28273, USA) was used to apply the load to the end of the trunk. Essentially a crank, the actuator pulled the trunk in a horizontal plane at a rate of six pounds/second. A load cell (LCCA-10K, Omega Engineering, Inc., One Omega Drive, Stamford, CT 06907-0047, USA) aligned with the pulling direction measured the load placed on the trunk. To measure the deflection of the trunk in the horizontal plane (x- and y- direction), two twenty-five-inch string pots (Rayelco P-25A, Ametek^® Aerospace, 1644 Whittier Ave, Costa Mesa, CA 92627-4115, USA ) were attached to the end of the trunk. A ten-inch string pot (Rayelco P-10A, Ametek^® Aerospace ) was also attached vertically above the end of the trunk to measure the movement in the z-direction. The data were recorded using the same data acquisition system and software as above.

CALCULATIONS OF MODULUS OF ELASTICITY

Coupons

The deflection at any location along the length of a simply supported beam can be determined using the following equation:

Equation 1

 

where:
Δ_y = deflection in the vertical direction (in)
P = load (lbs)
x = point distance from supports (in)
L = distance between supports (in)
E = modulus of elasticity (psi)
I = moment of inertia (in⁴)

For a rectangular cross section, the moment of inertia is:

Equation 2

b = width of coupon (in)
h = height of coupon (in)

The modulus of elasticity can be solved for by rearranging Equation 1.

Equation 3

 

The modulus of elasticity is constant over the linear portion of the stress-strain curve. To find this linear portion, the relative deflection, proportional to strain, was plotted against the relative load, proportional to stress, for each of the five points along the coupon (Figure 5). A linear section of the load vs. deflection curve was then chosen visually and a linear regression run using Microsoft Excel to verify its linearity. All sections used possessed R² values greater than or equal to 0.98. Equation 3 was then used to calculate the MOE along this linear portion, and these values were averaged to get a single MOE for each of the five positions along the coupon. These five values were again averaged to obtain one MOE for each coupon.

Figure 4. Deflected shape of dry coupon

Figure 5. Relative deflection versus relative load curves for green coupon

Figure 6. Relative displacement versus relative load for dried coupons

Trunks

The taper of the trunk complicates the calculation of the SMOE. The cantilever beam was separated into six-inch sections with decreasing cross-sectional areas, and the moments of inertia were calculated using Equation 4 below.
For a circular cross-section, the moment of inertia is equal to

Equation 4

where:

R = radius (in)

The compressive stress of green live oak tree wood is 2040 psi (Green et al. 1999). It was assumed that a bending stress of 1000 psi would still be in the elastic range. The corresponding load on the end of the cantilever beam was found from Equation 5 below. The load placed on the end of the beam was calculated to be 130 pounds.

Equation 5

where:

P = load (lbs)
I = moment of inertia (in⁴)
f_b = bending stress (psi)
L = distance from support to loading (in)
c = distance from neutral axis to extreme fiber (in)

MASTAN (Version 2.0), a structural analysis program, was used to help calculate the structural modulus of elasticity. A value of 1.00 x 10⁶ psi was assumed for the modulus of elasticity, and the deflection at the end of the trunk was then computed using the load of 130 pounds. To calculate the true SMOE, the assumed 1.00 x 10⁶ psi SMOE was multiplied by the ratio of calculated deflection to actual deflection. This number was then substituted back into MASTAN as the new modulus of elasticity. The process was repeated until the calculated deflection and the test deflection were equal. Because the difference between the deflection in the y-direction and the total deflection from all three directions was negligible, only the deflection in the direction of the pulling motion (y-direction) was used to find the SMOE.

RESULTS AND DISCUSSION

After the calculations for the MOE and SMOE were performed as described above, it was found that the data for both calculations were invalid. The values selected for the coupon MOE calculations appeared in a linear portion of the load-deflection graph. This slope of this linear, elastic region should yield a constant MOE, as in steel and concrete. However, the values only increased with time and with additional loading. To perform a statistical analysis using the mean as the test statistic, the data must meet three requirements: 1) independence of each other, 2) normal distribution, and 3) equal variance. The data did not meet the second and third criteria, and thus the analysis cannot be completed. No valid statistical conclusions can be drawn from the results of the coupon test.

A similar problem existed for the trunk data. The calculations were too complicated to allow for many different data points as in the coupon data. Each data point must be analyzed separately to find the SMOE, making the process very time-consuming. Instead of using the elastic portion of the load-deflection curve (Figures 7 and 8), one data point was selected to calculate the MOE. This data point was selected at 130 pounds, when the trunk was still assumed to be in the elastic range. This results in only one MOE value for each trunk. A sample size of one does not allow for statistically significant analysis.

Figure 7. Relative displacement versus relative load for green trunk

Figure 8. Relative displacement versus relative load for dried trunk

Though no final conclusions can be drawn about the data, observations can be made about the results of the experiment. These visual conclusions should be validated later with additional tests.

Coupons

It was assumed that the MOE of the upper and lower portions of the coupons would not be statistically different. While this was observed during the experiment, it cannot be stated that the MOE is constant along the trunk of young live oak trees. However, there are obvious visual differences between the green and dry coupons, as seen in the graphs below. It can be seen in Figures 3 and 4 that the deflection along the length of the green coupon is greater than the than that of the dried coupon. This agrees with the hypothesis that the dry coupons would be stiffer and therefore have a larger modulus of elasticity than green coupons. The dried coupons will deflect less before failure. Unfortunately, the magnitude of this difference is beyond the analytical parameters of this project.

There are also distinct differences in the load-deflection curves of the green and dried coupons (Figures 5 and 6). With the early loading, the green coupon deflected very little, but as the loading was increased, the deflection also increased. The green coupon deflects much more than the dried coupon. The dried coupon has a very pronounced point in the curve, signifying the failure of the sample at around 500 pounds. The green coupons did not behave in this way. There is no obvious failure point; the coupons instead were loaded until there was a visual crack in the coupon. The dried coupons were also able to sustain a greater load than the green coupons, indicating a stronger and stiffer material.

The data from one of the dried samples were actually thrown out; the calculations resulted in a negative value for the modulus of elasticity. This can partly be explained by errors in the test procedure. The dried coupons were tested first. The coupons were simply placed in the machine and loaded until failure. However, the green coupons were set up and then preloaded to help seat the sample. Preloading the samples helped minimize movement during the actual test.

Trunks

The bark of the tree is a softer material than the interior hardwood. Removing this softer outer layer should increase the structural modulus of elasticity, as stated in the hypothesis above. There were no noticeable differences seen throughout the test, and there cannot be any statistical evidence that bark significantly lowers the SMOE.

There are distinguishable variations in the load-displacement graphs for green and dried trunks. The green trunk shows a very smooth loading and unloading curve (Figure 7). The load is increased until a certain point and then decreased. There is no perceivable fracture in the graph. The dried trunk load-deflection graph (Figure 8) shows several sharp drop-off points. The load and deflection increase as with the green trunk, but then the load decreases suddenly. The points indicate the failure of the trunk, and they were accompanied by audible cracking noises during the tests. Mirroring the results of the green and dried coupons, the green trunk experienced more deflection before fracture than the dried coupon. Both trunk types were subjected to about the same load (around 1500 pounds) due to the limitation of the actuator. It could be reasonably inferred that the dried trunk would sustain a larger load than the green trunk.

As with the coupons, the test set-up did not allow for the trunks at the same point every time. Though it was attempted to cut all of the trunks to the same length, the physical properties of the trees did not allow for it. The trunks were cut just below the start of the lowest branch, and this height varied from tree to tree. The length of the trunk affects the placement of the actuator. The actuator did not always pull in a straight line, resulting in the load being applied at a different location than expected. Pulling the trunk at an angle instead of in a perpendicular line introduces minor errors into the deflection calculations.

One of the four goals of the research was to determine if the SMOE differed from the MOE. It is reasonable to conclude that the trunk would support a larger load than the coupon, simply due to the larger size. This does not denote that structural modulus of elasticity of the whole sample of the trunk actually varies from that of an interior sample. The data were unable to be compared using a mean as the test statistic. The data could perhaps be analyzed using more advance statistical methods beyond the scope of this research project.

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ACKNOWLEDGEMENTS

Special thanks to:
Dr. Perry S. Green
Dr. Thomas Sputo
Scott Jones
Chuck Broward
Danny Brown
John Gamache
University Scholars Program for providing funding

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REFERENCES

1. Cannell, M.G.R., and J. Morgan. (1987). “Young’s Modulus of Sections of Living Branches and Tree Trunks.” Tree Physiology 3, 355-364.
2. Green, D.W., J.E Winandy, and D.E. Kretschmann. (1999). “Mechanical Properties of Wood.” Wood handbook – Wood as an Engineering Material. Gen. Tech. Rep. FPL-GTR-113. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 463 p.
3. Morgan, J., and M.G.R. Cannell. (1987). “Structural Analysis of Tree Trunks and Branches: Tapered Cantilever Beams Subject to Large Deflections Under Complex Loading.” Tree Physiology 3, 365-374.
4. Niklas, Karl J. (1996). “Mechanical Properties of Black Locust (Robinia pseudoacacia) Wood: Correlations among Elastic and Rupture Moduli, Proportional Limit, and Tissue Density and Specific Gravity.” Annals of Botany 79, 479-485.
5. Niklas, Karl J. (1996). “Mechanical Properties of Black Locust Wood. Size- and Age Dependent Variations in Sap- and Heartwood.” Annals of Botany 79, 265-272.
6. Spatz, H.-C., and F. Bruechert. (2000). “Basic Biomechanics of Self Supporting Plants: Wind Loads and Gravitational Loads on a Norway Spruce Tree.” Forest Ecology and Management 135, 33-44.
7. “Standard Test Methods for Small Clear Specimens of Timber.” ASTM D 143-2000 (Reapproved 2000). Annual Book of ASTM Standards 2003, Vol. 4.10, 25-33.
8. “Standard Test Methods of Static Tests of Wood Poles.” ASTM D 1036-99. Annual Book of ASTM Standards 2003, Vol. 4.10, 123-134.

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