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					               University of Maine USDA FPL
             Co-operative Research Agreement
                Small Diameter Timber Study




Authors:
University of Maine:
     M. L. Peterson – Asst. Professor
     B. Woodward – Research Engineer
     L. Espinosa – NSF Research in Undergraduate Experience Student
     R. Urbina – Research Assistant
Applied Research Associates:
     J. Kainz – Research Engineer
I.     Motivation
         In the past, naturally occurring fires have cleared the forest ground cover from fuel
accumulation. These small, periodic fires reduce the amount of small diameter trees, shrubbery,
and fallen timber that cover the forest floor. These small naturally occurring fires are necessary
for some trees to release their seeds and do not harm the larger, more mature timber. However,
humans have suppressed fires within the last century in order to “save” the forest from damage,
and as a result, fuel has accumulated on the ground. Prescribed burns have been carried out to
remove the excess fuel on the forest floor, but unless care is taken, wildfire erupts like it did near
Los Alamos, New Mexico. The result was very destructive because the fire burned much more
fiercely and hotter as it fed upon nearly one hundred years of accumulated fuel. Thinning of the
forest is necessary to help prevent these disasters.
         Fire suppression results in a forest composed of tall, skinny trees of approximately the
same age which are competing for resources. Traditionally, the forest products industry in the
United States has exhibited a preference for large diameter timbers and little attention has been
paid to the use of smaller trees. However, forest stands are now overgrown with these small
diameter timbers (SDT) which pose a potential problem in their susceptibility to fire, insects, and
disease. One method of alleviating the problem is to remove these trees from the forests.
Unfortunately, there are no high-end applications for SDT. Small diameter timbers are often used
for fence posts, firewood and poles as well as chips for the production of manufactured wood
goods. In order to make thinning the forests economical, new, higher value markets must be
developed to provide incentive for harvesting and using these SDT.
         One suggested higher end use for SDT is in structural applications. However, there is a
lack of experience in how to accommodate their round, tapered shape, and the engineering
properties of SDT are not well known. In order to investigate this potential use, methods are
needed to assign allowable stresses with greater accuracy than is currently obtained with visual
inspection methods.
         Wood is a complex anisotropic material that is generally modeled as orthotropic. As a
tree develops, the demands on the supporting structure of wood change from fast, upward growth
in order to compete for sunlight to lateral strength to resist horizontal wind forces. These changes
result in variations that are much more complex than orthotropicity. Mechanical properties in any
one principal direction vary in the tree as a result of gradual transition from juvenile to mature
wood.
         In addition to naturally occurring material property variations, wood moisture content
also varies from pith to bark. Variations in the moisture content above the fiber saturation point
have little effect on the mechanical properties of wood, but the quantity of free water in the wood
can have significant effects on the dynamic response of wood. As the wood dries below the fiber
saturation point, its stiffness and strength properties change as well as its dynamic response.

        Ultrasonic waves have been used to determine elastic properties of materials. When an
SDT is excited by an external source of energy, the velocity and attenuation of the elastic waves
propagated in the timber are related to the mechanical properties of the wood. These
relationships are relatively well understood for isotropic, uniform materials but are complex and
less understood for wood. In order to assess the quality and condition of small diameter timber
using non-destructive methods, more work needs to be completed to assess the strength and
elastic properties of small diameter timbers.


                                                     I-2
II.    Destructive Testing
       A. Summary
        A total of 99 Ponderosa pine logs were broken in three point bending on an 8 foot span to
determine their flexural strength. Eighteen log specimens were from South Dakota, and eighty-
one were from the north rim of the Grand Canyon. South Dakota logs ranged from 5 ¾” to 8 ¾”
in diameter with an 8 3/8” diameter log having the maximum failure strength of 17,800 lbs.
Grand Canyon logs ranged from 3 ½” to 8 ½” in diameter with 7 ½” log having a maximum
failure strength of 14,160 lbs. Experiments were conducted using ½” chain covered with fire
hose for supports and loaded at a rate of 0.125 in/sec as described in the procedure section
below.

       B. Procedure
        Logs were simply supported at the ends using a loop of ½” chain bolted to the test frame
structure. The load was applied by another loop of chain attached to the 50,000 lb. actuator
located at the center of the 8 foot test span. At least 5 inches of overhang were allowed for the
logs to ensure that the specimen did not slip out of the supports when loaded. End supports and
load chains were covered with a sleeve of fire hose to prevent damage to the test specimens.
Figures 2.1 and 2.2 show the test set-up before and after testing.
        The load was applied by moving the MTS actuator head upward at a rate of 0.0125
in/sec. A maximum of 6 inches of throw was available for the actuator movement, and most
specimens broke within this range. Logs were considered to have failed when the load dropped
below 1,000 lbs. Data was collected on log diameters, overall length, failure load, time to failure,
moisture content, and visible defects around the failure site.




                           Figure 2.1 Test set-up prior to log breaking.




                                                    II-3
                             Figure 2.2 Test set-up after specimen failure.




                                Figure 2.3 Log 18C at the fracture point.

        Additionally, after the logs had been tested in 3 point bending, one inch cube samples
were harvested near the fracture site. The fracture site was removed along with several inches of
undamaged wood on either side of the break point using a horizontal band saw. A 1 ½ - 3” thick
disk was then cut from this log section as close to the fracture point as possible while still
collecting undamaged wood.
       C. Results
        Table 1 shows the dimensions and strengths of the logs tested. South Dakota (SD) log
samples are denoted by a “C” after the log number. All other samples are from the Grand
Canyon (GC) test group. All samples broke in the middle of the test span where the chain was
applying the load (see Figure 2.2).
        Four of the GC test group required flipping before they broke. When the actuator had
traveled its entire 6” throw and the sample had not failed, the log was rolled 180 degrees about
its central axis and reloaded. These logs are denoted in the table by two lines of data.
Additionally, GC logs 1, 2, 3, 9 and 14 have no failure load due to technical difficulties with the
test machinery. Figure 2.3 shows a test specimen that has broken.


                                                    II-4
        Table 1.1 Log breaking data for samples tested. Log diameters are given for both ends of
the log since most logs involved a significant taper through the 8 ft. section.

      Log Overall      Span Length      Diameter (in)      Load at   Time to
      #   Length       (in)                                Failure   fail
          (in)
                       A       B        A       B          (lbs.)    (minute
                                                                     s)

      South Dakota Logs

      3C    118        47      47       6.5     7.25       8850      11.80
      12C   125.25     46      48       6.75    5.625      4620      6.58
      14C   99         42.5    42       7.25    6.75       9450      7.92
      15C   97.5       42      43       6.5     5.75       7680      6.60
      17C   150.25     49      48       7.75    6.875      6070      5.60
      18C   137        47      47       6.25    7.125      9984      8.05
      19C   141.6      48      48.5     7.625   6          7920      6.82
      20C   129        47      48       8.0     7.0        10900     6.72
      21C   130        47      47       7.0     7.375      9035      3.93
      22C   123.75     47.5    47.5     6.75    7.857      9950      6.03
      23C   136        47      49       7       7.25       6550      6.00
      24C   153        47      47       6.875   7.375      9581      5.15
      25C   119.75     47      49       7.125   6          9440      7.9
      26C   140        47      48       8.75    8.5        9450      5.03
      27C   103.5      44      46       7.25    6.75       13800     9.03
      28C   139.4      47      48       8       7.25       10800     6.17
      29C   127        47      48       8.375   8.125      17800     10.13
      30C   137        47      47       6.875   7.325      9350      7.73

      Grand Canyon Logs

      1     99.5       42.5    43.5     5.25    3.75       -         0.00
      2     99         43      42       4.25    4.5        -         0.00
      3     100        41      42       4       4.25       -         0.00
      4     99         42      43       4.5     4.5        4385      14.23
      5     96         43      42       6       5          3518      6.47
      6     99         43      42       7.5     7          13915     17.05
                                                           11543     11.10
      7     94         42      42       5.25    4.5        4693      14.32
      8     97         42      44       6       6          5032      5.03
      9     100        45      42       6       7          -         0.00
      10    100        43      42       8       7          9727      7.90
      11    97         44      42       8       8          6862      5.45
      12    100        45      44       5       5          4769      17.05




                                                    II-5
 Table 1.1 (cont.)

Log Overall      Span Length   Diameter (in)      Load at   Time to
#   Length       (in)                             Failure   Fail
    (in)
                 A      B      A       B          (lbs.)    (minute
                                                            s)

                                                  3573      15.78
13   97          44     43     6       5          5236      9.25
14   96          43     43     5.5     4.5        -         0.00
15   95          42     43     4.5     5          3231      6.58
16   99          42     43     4.5     5          3057      7.30
17   97          44     42     4.5     5          2960      9.03
18   99          44     42     8       6          7607      6.42
19   96          42.5   42     6.5     5.5        6504      7.78
20   99          45     42     7       0.5        9819      7.23
21   98          43     41     6.5     7.5        10974     10.50
22   99          43     42     6       8          10186     11.28
23   97.5        43     42     7       8          7354      6.90
24   101         44     43     5       6          4630      7.98
25   99          43     41     7       7          12951     10.03
26   98          43     43     4.5     4          2465      11.32
27   99          43     43     5       5          3427      8.75
28   99          43     42     4       5          2800      9.97
29   99          43     43     6       4          3756      7.43
30   100         43     43     4       4.5        2616      8.10
31   97          43     43     5.5     5          4333      8.95
32   97          43     42     6       4          3397      6.03
33   98          43     41     5.5     6          5565      5.88
34   99          42     40     6       7.5        14160     9.63
35   98          44     42     7       8          10764     9.20
36   98          41     45     6       7.5        12752     7.85
37   99          43     43     7       7          8618      7.35
38   99          42     45     7       6          8952      6.40
39   99          42     46     6       5.5        7247      7.62
40   100.5       44     45     5       5          4688      10.55
41   100         42     44     4       4          3273      16.82
42   99          43     44     4.5     4          2048      17.05
43   99          44     43     5       4.5        3812      13.25
44   97          43     41     4       5          4222      14.75
45   95          43     41     4       4.5        2585      10.50
46   100         43     43     4       3.5        1534      11.52
47   99          43     44     4.5     5          2615      7.80
48   98          45     43     5       6.5        4883      8.13
49   100         42     44     5.5     4          4098      15.67
50   98          42     43     7       7          6472      6.35
51   100         44     43     7       6          5953      7.48


                                           II-6
      Log Overall      Span Length       Diameter (in)      Load at    Time to
      #   Length       (in)                                 Failure    Fail
          (in)
                       A        B        A       B          (lbs.)     (minute
                                                                       s)
      52    98         42       44       6.5     7          8445       17.05
                                                            6638       17.05
      53    99         43       43       7       8          7772       7.27
      54    98         43       43       8       7          8225       7.90
      55    99         42       44       7       7          12838      10.72
      56    99         42       42       4       4          1763       9.28
      57    99         43       42       4       4          1656       12.50
      58    98         41       42       4.5     3.5        2235       10.20
      59    99         43       42       4       4.5        2703       8.30
      60    100        42       43       5       5          3102       9.40
      61    99         43       42       8.5     7.5        11238      14.17
      62    95         42       43       4       5          4005       8.87
      63    99         43       44       4       5          2458       9.08
      64    97         45       41       4.5     4.5        3516       13.05
      65    99         44       43       4.5     5.5        4070       13.15
      66    98         44       42       5       4          3318       10.60
      67    99         43       44       5       5          4783       9.50
      68    100        41       44       4       5          4785       17.05
      69    99         41       45       3.5     6          3985       17.05
                                                            2585       11.08
      70    97         42       44       6.5     5          5981       9.08
      71    99         41       44       5       4          4833       17.02
      72    97         41       43       5       4.5        4074       11.22
      73    100        43       41       7       8.5        10308      7.83
      74    97         45       42       8.5     7.5        11860      8.03
      75    94         40       43       6/5     7          5322       13.17
      76    97         42       44       6       6          3873       5.48
      77    100        43       45       5       5          4395       9.95
      78    100        43       44       4.5     4          4104       10.20
      79    100        42       44       4.5     4.5        3416       12.98
      80    105        44       46       5       4          2041       12.42
      81    99         44       42       4       3.5        1320       8.82


        Figures 2.4 and 2.5 show typical load versus displacement graphs for the flexural strength
tests. Small drops in the graph indicate points where the sample started breaking but complete
failure followed at a later time and significantly higher load. Complete failure was defined by the
applied load dropping below 1,000 lbs.




                                                     II-7
                                                               Log 18C

                              12000

                              10000

                              8000
              Load (lb)


                              6000

                              4000

                              2000

                                  0
                                  0.00    0.50          1.00     1.50        2.00    2.50          3.00   3.50
                                                                Displacement (in)


                 Figure 2.4 Load versus displacement curve for South Dakota log 18C.




                                                               Log CO64


                              4000
                              3500
                              3000
                              2500
                  Load (lb)




                              2000
                              1500
                              1000
                               500
                                  0
                               -5000.00          1.00          2.00           3.00          4.00          5.00

                                                               Displacement (in)



           Figure 2.5 Load versus displacement curve for Grand Canyon log 64.




        Figures 2.6 and 2.7 show the load at failure versus the average diameter of the South
Dakota and Grand Canyon logs respectively. Figures 2.8 and 2.9 show the load versus time to
failure graphs for the two experimental sets.




                                                                      II-8
                                      Load vs. Avg. Diameter - SD Logs


    Avg. Diameter (in)   10.00

                          8.00

                          6.00
                                                                            2
                                                                        R = 0.2811
                          4.00

                          2.00

                          0.00
                                  0        5000           10000          15000           20000
                                                        Load (lb)



                         Figure 2.6 Load versus average log diameter for South Dakota logs




                                      Load vs. Avg. Diameter - GC Logs
Avg. Diameter (in)




                         10
                          8
                          6
                          4                                                     2
                                                                            R = 0.6677
                          2
                          0
                              0               5000                  10000                15000
                                                       Load (lb)

Figure 2.7 Failure load versus average log diameter for Grand Canyon logs.




                                                           II-9
                                                  Load vs. Time to Failure - SD Logs

                                      14.00
                                      12.00
          Time (minutes)


                                      10.00
                                       8.00
                                       6.00
                                       4.00
                                       2.00
                                       0.00
                                              0           5000           10000            15000           20000
                                                                        Load (lb)

                                          Figure 2.8 Load versus time to failure for South Dakota logs.



                                                  Load vs. Time to Failure - GC Logs
          Time to Failure (minutes)




                                      20.00

                                      15.00

                                      10.00

                                       5.00

                                       0.00
                                              0   2000    4000   6000     8000      10000 12000 14000 16000
                                                                        Load (lb)

                                         Figure 2.9 Load versus time to failure for Grand Canyon logs.


       Figures 2.10 and 2.11 show the percentage of mature wood to the failure load for the
South Dakota and Grand Canyon log sets. Percentage of mature wood was determined by
measuring the juvenile wood in the sample and then subtracting that from the total wood in the
sample. Juvenile wood was determined to be the wood encompassed by the first 20 growth rings.
The radius of the first 20 growth rings was measured starting from the pith, and the log diameter
was estimated along the same radial path.




                                                                         II-10
                                 Failure Load vs. % Mature Wood - SD Logs

                      0.60
   % Mature Wood

                      0.50
                      0.40                                               2
                                                                        R = 0.2444
                      0.30
                      0.20
                      0.10
                      0.00
                             0            5000           10000          15000          20000
                                                        Load (lb.)



Figure 2.10 Percentage of mature wood versus failure load for South Dakota logs.



                                     Load vs. % Mature Wood - GC Logs
          % Mature Wood




                          0.80
                          0.60
                          0.40                                                   2
                                                                                R = 0.1946
                          0.20
                          0.00
                                 0               5000                10000             15000
                                                        Load (lb.)


Figure 2.11 Percentage of mature wood versus failure load for Grand Canyon logs.




                                                           II-11
        It is hypothesized that the strength of the log correlates to the amount of mature wood
that supports the load. The moment of inertia of the mature wood portion of the beam would be
expected to be related to the strength. The moment of inertia for each log was calculated
assuming a hollow cylinder. The effect of the load carrying capacity of the inner cylinder of
juvenile wood was assumed to be negligible. The moment of inertia is then simply given by the
relationship:
                                     D 4  d 4 
                                 I                                           (1)
                                         64

Log    Diameter Juvenile Wood     Total Log Diameter (in.)    Moment of Inertia
       (in.)                                                  (in4)
South Dakota Logs
3C     3.50                       6.25                        67.50
30C 4.13                          6.88                        95.40
26C 4.50                          7.38                        125.02
17C 4.63                          7.13                        103.99
29C 3.38                          7.63                        159.48
23C 3.88                          6.88                        98.55
28C 4.38                          7.50                        137.26
20C 5.00                          8.00                        170.30
25C 4.13                          6.13                        54.85
27C 3.25                          6.88                        104.13
19C 2.88                          6.50                        84.23
12C 3.50                          5.63                        41.76
22C 3.00                          7.00                        113.83
14C 3.88                          7.13                        115.38
15C 4.38                          5.75                        35.66
21C 4.50                          8.13                        193.70
24C 3.88                          6.38                        69.97
18C 3.75                          6.63                        84.81

Grand Canyon Logs
4     2.63                        4.50                        17.79
5     4.00                        5.50                        32.33
7     3.13                        5.88                        53.77
8     4.00                        5.75                        41.07
10    5.13                        7.63                        132.00
11    4.38                        7.25                        117.58
13    3.63                        5.25                        28.80
15    3.50                        5.25                        29.91
16    3.88                        5.88                        47.39
17    4.00                        5.38                        28.39
18    3.00                        7.13                        122.47
19    5.13                        7.13                        92.59
20    5.00                        8.50                        225.45
21    3.63                        5.38                        32.48
23    5.25                        7.88                        151.42


                                                 II-12
Log    Diameter Juvenile Wood     Total Log Diameter (in.)   Moment of Inertia
       (in.)                                                 (in4)
24     3.63                       6.00                       55.11
25     3.50                       7.00                       110.44
26     3.50                       4.88                       20.35
27     3.50                       5.00                       23.30
28     3.13                       5.50                       40.22
29     3.13                       5.38                       36.27
30     5.25                       6.38                       43.76
31     3.38                       5.13                       27.48
32     2.75                       5.75                       50.83
33     3.75                       7.25                       125.85
34     3.13                       7.50                       150.56
35     3.25                       7.63                       160.37
36     4.63                       9.25                       336.73
37     4.25                       8.38                       225.37
38     4.63                       7.38                       122.69
39     3.50                       6.13                       61.69
40     3.88                       5.50                       33.83
41     2.88                       4.63                       19.10
43     2.50                       5.38                       39.03
44     2.75                       4.75                       22.17
47     3.25                       5.00                       25.19
48     3.50                       6.13                       61.69
53     4.38                       7.88                       170.72
54     5.38                       7.75                       136.04
55     3.13                       7.13                       121.76
57     3.25                       4.13                       8.73
59     2.50                       4.38                       16.06
60     2.75                       4.75                       22.17
61     5.00                       8.38                       210.71
63     3.50                       4.88                       20.35
65     3.25                       5.25                       31.80
67     3.00                       5.88                       54.48
68     2.75                       6.00                       60.78
70     3.38                       5.25                       30.91
71     3.25                       5.63                       43.64
72     3.25                       4.75                       19.50
73     3.50                       7.50                       147.87
74     4.63                       9.63                       398.62
75     3.75                       7.00                       108.10
76     3.63                       5.50                       36.42
78     2.75                       4.63                       19.64
79     3.13                       5.13                       29.17
80     2.50                       5.50                       42.98

       Table 2. Calculated moments of inertia and estimated juvenile wood and total log
diameters for South Dakota and Grand Canyon logs.


                                                 II-13
where D refers to the total diameter of the log and d refers to the diameter of the juvenile wood
 (based on measurments of the first 20 annual growth rings). Table 2 gives the calculated
moments along with the corresponding juvenile wood and total log diameters.
        Figures 2.12 and 2.13 show the moment of inertia for the mature wood cylinder versus
the load at failure for the two experimental log sets. Figure 2.14 shows the moment of inertia
versus the failure load for the GC logs whose failure strength was under 10,000 lbs.




                                  Moment of Inertia vs Failure Load - SD logs

                        250
    Moment of Inertia




                                                                          2
                                                                         R = 0.2453
                        200
                        150
                        100
                        50
                         0
                              0            5000         10000                 15000     20000
                                                  Load at Failure (lb)

Figure 2.12 Moment of inertia for mature wood cylinder versus load at failure for South Dakota
                                          log set.




                                                         II-14
                                   Moment of Inertia vs. Failure Load - GC Logs

                         500
     Moment of Inertia



                         400                                               2
                                                                      R = 0.5945
                         300
                         200
                         100
                          0
                               0      2000     4000     6000       8000        10000 12000 14000 16000
                                                        Load at Failure (lb)

 Figure 2.13 Moment of inertia for hollow mature wood cylinder versus failure load for Grand
                                        Canyon logs.




                                   Moment of Inertia vs. Failure Load - GC Logs

                         250
                                                               2
                                                             R = 0.7288
     Moment of Inertia




                         200
                         150
                         100
                         50
                          0
                         -50 0          2000          4000         6000          8000    10000   12000

                                                        Load at Failure (lb)

Figure 2.14 Moment of inertia for hollow mature wood cylinder versus load at failure for Grand
                  Canyon logs with failure strengths less than 10,000 lbs.

        Taking the same sample set used in Figure 2.14 and comparing the moment of inertia and
the load at failure from the Grand Canyon logs, a correlation coefficient of 0.85 was calculated.


                                                                   II-15
This value indicates a strong correlation since the correlation coefficient is near 1.0. In order to
determine if the sample is representative of the population, the probability was evaluated using a
Student’s t-test. A calculated percentile, t, was calculated by

                                                      r n2
                                                 t
                                                        1 r2

        where r is the sample correlation coefficient and n is the sample size. The calculated
value, t = 11.24, was compared to the Student’s t distribution at 0.05 and 0.01 significance
intervals. Since the calculated value t was larger than percentile values at t95 and t99, the null
hypothesis can be rejected and it is assumed that the sample correlation coefficient is
representative of the population.
        Moisture readings were taken using a hand held moisture meter calibrated for Douglas
fir. Care was taken to place the probes of the unit within a single growth ring to maintain
accuracy of the measurements. Measurements were taken in both the juvenile and mature wood
portions of the freshly cut log ends after the fracture site was harvested. A moisture measurement
was also taken along the longitudinal length of the specimens. Overall the moisture content was
quite low for the South Dakota log group. Values ranged from 6- 11.5%. Moisture content for
the Grand Canyon logs was assumed to be less than 6% as these logs had previously been the
subject of a drying study (Ref. ?). Table 3 shows typical moisture readings.


Log #     Cut end -     Cut end –        Longitudinal         Fracture site -mature   Fracture site -
          mature        juvenile                                                      juvenile
18C       6.75          7.875            7.25                 6.00                    6.00
24C       7.625         8.125            8.75                 7.25                    8.50
21C       7.5           7.75             11.5                 6                       6.25
3C        6.75          7                7.5                  6.75                    7.25
30C       7.25          7.5              9.0                  9.375                   11.5
26C       6.25          7                7.25                 7.0                     7.25
17C       7.0           7.0              7.5                  6.5                     6.5
29C       6.75          7.5              8.25                 6.75                    7.25
23C       7.75          8.125            7.5                  7.0                     7.375
28C       7.375         7.75             8.5                  7.25                    7.625
20C       6.0           6.5              7.75                 9.25                    8.875
25C       7.75          7.75             8.5                  7.75                    7.25
27C       7.0           7.625            8.5                  7.25                    7.0
19C       7.75          8.5              8.5                  7.25                    7.25
12C       6.75          6.25             6.25                 7.25                    7.0
22C       6.25          6.5              8.5                  7.0                     6.0
14C       6.75          7.0              7.125                6.25                    7.5
15C       6.0           6.5              7.0                  6.5                     6.5

        Table 3. Moisture content of logs.




                                                      II-16
       D. Conclusions
        All log samples in both test groups broke in the middle of the test specimen where the
load was applied regardless of visible defects that were evident along the length of the log. This
suggests that defects such as knots and cracks are not critical to the strength of the log. The
moisture content of the samples tested was reasonably consistent and was held below 12% for all
cases. Hence, varying moisture content could not explain the differences in log strength for
similarly sized logs.
        Figures 2.8 and 2.9 show no correlation between the time to failure and the strength of
the log. This suggests that the load rate of the sample was sufficient to keep creep from affecting
experimental results.
        Figures 2.6 and 2.7 indicate that the diameter of a log alone is not necessarily an indicator
of the log strength. The smaller logs in the Grand Canyon sample group suggest that as the
diameter of the log increases so does the failure strength of the log. Figures 2.12 and 2.13
suggests that while the percentage of mature wood alone in the sample does not predict is
strength (figures 2.10 and 2.11), the strength of a log is closely related to the moment of inertia
of the mature wood in the log. The juvenile wood was subtracted from the volume of wood in the
cylindrical timber and the moment of inertia was calculated for the hollow cylinder of mature
wood. For the GC log set, the moment is clearly related to the failure strength of the log up to a
failure strength of 10,000 lbs. (see figure 2.14). Above this failure load, the logs were subject to
some crushing of the fibers and the correlation is not as strong. This suggests if one is able to
non-destructively determine the percentage of mature wood in a tree, he could estimate the
timber’s moment of inertia and predict the tree’s strength before harvesting.




                                                   II-17
III.    Material Properties
       A. Summary
        Seventeen juvenile and twenty mature wood specimens were ultrasonically tested to
calculate the portion of the elastic tensor that could be determined using only longitudinal waves.
Specimens were tested in all three of the geometric axes: longitudinal, tangential, and radial.
Cross-correlation between signals in a reference aluminum cube and the test wood samples was
preformed to determine the relative time delay and velocity of the waves propagating through the
samples. The stiffness elasticity tensor was obtained from standard calculations after windowing.
Comparisons were then made between the mature and juvenile wood samples. In the tangential
and radial directions, the phase velocities (V22, V33) and elastic constants (C22, C33) are similar for
mature and juvenile wood samples. However, the phase velocity (V11) and elasticity constant
(C11) of mature wood in the longitudinal direction is greater than that of juvenile wood.

       B. Procedure
        After the log specimens were loaded to failure in three point bending, one inch cubes
were harvested from the area near the break. A 1 ½ - 3” disk were removed from the log using a
horizontal band saw. From these disks, one inch cube samples were harvested from both the
juvenile and mature wood regions of the disk. The first 20 rings on the log were designated as
juvenile wood (see figure 3.1). Samples were collected from the neutral axis of the disk so that
the rings were of uniform direction across the sample surface (figure 3.2).



                                                                                Juvenile
                                                                                Wood




                                                                                 Mature
                                                                                 Wood




               Figure 3.1 Cross section of log showing juvenile and mature wood regions.




                                                    III-18
                                                                       Log Name
                                                                            S


                                                                       Log Name
                                                                            H




           Figure 3.2 Position of the cubes harvested from the neutral axis of the disk.


        In addition to the normal orientation, cubes were harvested at 30, 45, and 60 degree
angles from the axis of the disk (figure 3).


                     Log Name                                      Log Name
                       45 H                                          60 S




                                                                Log Name
                                                                  30 S




   Figure 3.3 Position of cubes harvested of 30, 45, and 60 degrees from the axis of the disk.

        For several log specimens, additional 5” cylinders were cut. Cylinders were cut into four
parts, cutting a cross through the center of the log section in the longitudinal axis. Cubes were
then harvested from each quadrant to harvest cubes with different orientations to the longitudinal
axis of the log (figure 3.4). Cube specimens were polished on a belt sander to obtain flat, smooth


                                                  III-19
faces and ensure parallel sides. Cubes were then labeled and stored in ziplock bags in a
refrigerator at 48 degrees F with 37% humidity to maintain a consistent moisture content until
ultrasonic testing.



                         Log Name
                          LR 45 S
   Radial
                                                  Log Name
                                                   LR 60 S




                                                                  Log Name
                                                                   LR 30 H




                                                                                      Longitudinal


   Figure 3.4 Positions of the cubes harvested at a range of angles from the longitudinal axis.

        Dimensions and densities of the juvenile and mature wood samples were recorded, and
each sample was tested ultrasonically. The testing apparatus consisted of square wave pulser
which excited an ultrasonic transducer (Panametrics V194) to propagate an elastic pulse through
the wood cube samples. A receiving transducer is then excited by the wave signal which has
propagated through the cube. The signal is converted into a small voltage which runs through an
ultrasonic amplifier (Panametrics Ultrasonics Preamp at 60 dB) connected to a digital
oscilloscope (Tektronix TDS 520A) which then displays the data as a wave. Waves were
recorded with the transducer in direct contact with the wood sample. A longitudinal wave was
generated with the sample oriented with the wave perpendicular to each of the three principal
axes: longitudinal, tangential, and radial. Signals were truncated using a trapezoidal window to
prevent multiple reflections in the time signal. The window function was defined by

                                                   1       t  ( 0.8 p )
                                      
                             w( t )  ( p  t ) /( 1  0.8 )       ( 0.8  t  p )
                                                             (t  p )
                                                     0

        where p is the flight time corresponding to the wave passing through the specimen twice
and t is time.

        Initial estimates were used for the phase velocity in order to position the time window.
The initial time estimates were as follows (m/s):



                                                      III-20
        V11 ' = 3800       V 22 ' =1700       V 33 ' =2300        (for the juvenile wood)

                 V11 ' = 5000      V22' = 1400         V 33 ' = 2300        (for the mature wood)

       Cross-correlation with a reference aluminum cube was used to determine the relative
phase velocity and relative time delay of the signal passing through the wood sample. The elastic
tensor was then calculated using the phase velocities and densities of the wood samples
according to the following equation:
                           C11  V11 2  , longitudinal direction

                            C 22  V22 2  , tangential direction

                                    C 33  V33 2  ,     radial direction

         For a more complete explanation of the analysis, cross-correlation, and calculation of the
elastic tensor, see Appendix A.
       C. Results
       Tables 1 and 2 show the dimensions, weights, and densities of each the harvested mature
and juvenile cube samples prior to ultrasonic testing.

Log      Length (cm)      Length     Length      Weight          Density
Name     longitudinal     (cm)       (cm) radial                 (g/cm3)
                          tangential

73A      2.581            2.644       2.624            0.020     0.507763
66B      2.621            2.634       2.642            0.018     0.448598
69A      2.604            2.616       2.606            0.020     0.512151
24A      2.591            2.578       2.616            0.016     0.416191
62B      2.512            2.573       2.548            0.018     0.496869
49B      2.487            2.504       2.553            0.014     0.400293
63A      2.568            2.570       2.619            0.014     0.368141
60B      2.601            2.637       2.616            0.018     0.456052
76B      2.578            2.550       2.565            0.014     0.377297
64B      2.520            2.502       2.586            0.016     0.44617
64B      2.545            2.593       2.558            0.018     0.484647
45A      2.578            2.586       2.606            0.016     0.418634
49A      2.537            2.499       2.626            0.016     0.436631
63A      2.583            2.606       2.642            0.016     0.408973
42A      2.565            2.520       2.611            0.016     0.430894
56A      2.548            2.631       2.644            0.014     0.358999
60B      2.647            2.662       2.644            0.018     0.439207
66B      2.573            2.604       2.596            0.020     0.522784



                                                        III-21
Log     Length (cm)    Length     Length      Weight    Density
Name    longitudinal   (cm)       (cm) radial           (g/cm3)
                       tangential

45A     2.510          2.560      2.553     0.014       0.38799
72A     2.548          2.573      2.550     0.016       0.435063
50B     2.604          2.614      2.621     0.014       0.356767
78B     2.609          2.548      2.654     0.018       0.463832
51B     2.644          2.611      2.619     0.016       0.402248
43B     2.548          2.560      2.637     0.016       0.4229
62A     2.530          2.568      2.578     0.016       0.43423
22B     2.502          2.489      2.535     0.014       0.403099
65B     2.593          2.570      2.593     0.018       0.473278
25A     2.601          2.598      2.596     0.018       0.466361
51A     2.578          2.504      2.563     0.014       0.384565
                                  Average Density       0.433125


Table 1. Dimensions, weights, and densities of mature wood samples.

Log     Length (cm)    Length     Length      Weight    Density
Name    longitudinal   (cm)       (cm) radial           (g/cm3)
                       tangential
50B     2.479          2.565      2.588     0.016       0.441825
76B     2.591          2.593      2.626     0.020       0.515181
25A     2.573          2.616      2.601     0.020       0.51923
69A     2.586          2.642      2.634     0.018       0.454767
45A     2.578          2.573      2.555     0.018       0.482698
51A     2.558          2.545      2.578     0.020       0.54168
24A     2.647          2.573      2.545     0.016       0.419615
50A     2.576          2.626      2.631     0.018       0.459653
22A     2.593          2.616      2.583     0.018       0.466836
51B     2.583          2.637      2.644     0.018       0.454339
22B     2.517          2.512      2.545     0.020       0.564896
76A     2.540          2.558      2.550     0.028       0.768194
43B     2.565          2.588      2.720     0.016       0.402635
65B     2.609          2.586      2.621     0.018       0.462755
49A     2.545          2.568      2.591     0.016       0.429514
78B     2.570          2.570      2.713     0.024       0.608632
21A     2.581          2.611      2.593     0.022       0.572251
                                  Average Density       0.503806
       Table 2. Dimensions, weights, and densities of juvenile wood samples.


                                               III-22
        Tables 3 and 4 show the phase velocities (V11, V22, V33) and elastic constants (C11, C22,
C33) for the mature and juvenile wood samples. V11 and C11 correspond to the longitudinal
direction while V22 and C22 refer to the tangential direction and V33 and C33 to the radial
component. Table 5 shows the average phase velocities for the juvenile and mature wood
samples in the longitudinal, tangential, and radial directions.

   Log          v11 ( m / s ) c11 ( N / m 2 ) v 22 ( m / s )      c22 ( N / m 2 ) v 33 ( m / s )   c33 ( N / m 2 )
   Name
                1  10 3      1.0  10 10     1  10 3            1.0  10 9      1  10 3         1.0  10 9

   73A         3.7836         0.7260           1.7557             1.5632           2.6448          3.5476
   66B         5.1798         1.2022           1.3236             0.7849           2.1616          2.0936
   69A         5.3344         1.4556           1.2553             0.8061           2.0359          2.1201
   24A         5.0596         1.0641           1.2253             0.6241           2.1585          1.9367
   62B         5.2991         1.3935           1.0748             0.5732           1.7354          1.4945
   49B         4.5044         0.8112           0.9883             0.3905           1.4687          0.8624
   63A         4.4118         0.7157           1.0273             0.3881           1.7862          1.1732
   60B         5.0206         1.1482           1.1759             0.6299           2.1947          2.1940
   76B         4.7915         0.8652           1.2776             0.6151           1.6033          0.9687
   64B         4.9594         1.0961           1.1341             0.5732           2.0852          1.9376
   64B         5.7574         1.6045           1.3381             0.8667           2.1208          2.1772
   45A         4.9195         1.0119           1.2638             0.6678           1.5794          1.0430
   49A         5.0142         1.0965           1.1657             0.5926           2.1145          1.9499
   63A         4.9105         0.9850           1.5151             0.9377           1.6468          1.1078
   42A         5.8297         1.4627           1.2611             0.6844           2.0462          1.8020
   56A         4.2315         0.6420           1.3039             0.6097           2.3906          2.0492
   60B         5.7781         1.4646           1.0759             0.5078           2.1496          2.0271
   66B         4.0968         0.8764           1.2234             0.7815           2.1524          2.4190
   45A         4.9786         0.9606           1.2227             0.5793           2.1378          1.7711
   72A         3.9312         0.6715           1.0308             0.4618           1.4440          0.9060

Table 3. Phase velocities and elastic constants for mature wood samples.




                                                         III-23
   Sample      v11 ( m / s ) c11 ( N / m 2 ) v 22 ( m / s )   c22 ( N / m 2 ) v 33 ( m / s )   c33 ( N / m 2 )
   Number
               1  10 3      1.0  10 9      1  10 3         1.0  10 9      1  10 3         1.0  10 9

   50B         3.3772       5.0330            1.5417          1.0488          2.3108           2.3565
   76B         3.5101       6.3400            1.8901          1.8383          2.1740           2.4320
   25A         4.1231       8.8160            1.9012          1.8746          1.8290           1.7349
   69A         4.5519       9.4110            1.3519          0.8301          2.0513           1.9113
   45A         3.6201       6.3200            2.0816          2.0891          2.1617           2.2529
   51A         3.4496       6.4280            1.6944          1.5533          2.1617           3.2738
   24A         4.5315       8.6060            1.4595          0.9378          2.1617           1.8788
   50A         3.5376       5.7450            1.6353          1.2277          2.4052           2.6559
   22A         4.0017       7.4660            1.7968          1.5053          2.3440           2.5618
   51B         4.2764       8.2980            1.1257          1.0966          1.5499           1.0900
   22B         3.8251       8.2550            1.4881          1.2495          2.2015           2.7346
   76A         2.5297       4.9100            2.4266          4.5180          2.3438           4.2148
   43B         4.5003       8.1440            2.3067          2.1398          2.3410           2.2038
   65B         3.1024       4.5640            2.1404          2.1175          2.6856           3.3335
   49A         4.4181       8.3730            1.8447          1.4599          1.5933           1.0891
   78B         2.5963       4.0970            2.0271          2.4980          2.2162           2.9857
   21A         2.9457       4.9590            1.1605          0.7697          2.5624           3.7530

                Table 4. Phase velocities and elastic constants for juvenile wood.



Mature Wood                                         Juvenile Wood
Direction      Velocity m/s      Std. Dev.          Direction   Velocity                 Std. Dev.
                                                                m/s
Longitudinal   4889.6            +/- 587            Longitudina 3699.9                   +/- 659
                                                    l
Tangential     1231.9            +/- 176            Tangential 1757.2                    +/- 376
Radial         1982.8            +/- 318            Radial      2181.9                   +/- 299

Table 5. Comparison of average phase velocities and standard deviations of juvenile and mature
                                      wood samples.


       Figures 3.5, 3.6, and 3.7 graphically show a comparison of the phase velocities in
juvenile and mature wood sample in the longitudinal, tangential, and radial directions.


                                                        III-24
                        Velocity(V11) in longitudinal direction

                    5                                                8
                                                                     7
                    4
        Frequency



                                                                     6
                    3                                                5
                                                                                 Juvenile
                                                                     4
                    2                                                3           Mature

                    1                                                2
                                                                     1
                    0                                                0
                       00
                       00
                       00
                       00
                       00
                       00
                       00
                       00
                       00
                        0
                        0
                     30
                     90
                    15
                    21
                    27
                    33
                    39
                    45
                    51
                    57
                    63
                                  Velocity(m/s)



    Figure 3.5 Comparison of juvenile and mature wood phase velocities in the longitudinal
                                         direction.




                        Velocity(V22) in tangential direction

                    6                                              12
                    5                                              10
        Frequency




                    4                                              8
                                                                                Juvenile
                    3                                              6
                                                                                Mature
                    2                                              4
                    1                                              2
                    0                                              0
                       00
                       00
                       00
                       00
                       00
                       00
                       00
                       00
                       00
                        0
                        0
                     30
                     90
                    15
                    21
                    27
                    33
                    39
                    45
                    51
                    57
                    63




                                 Velocity(m/s)



Figure 3.6 Comparison of juvenile and mature wood phase velocities in the tangential direction.




                                                III-25
                              Velocity(V33) in radial direction

                       10                                            10
                        8                                            8
           Frequency



                        6                                            6            Juvenile
                        4                                            4            Mature

                        2                                            2
                        0                                            0
                                 00

                                 00

                                 00

                                 00

                                 00

                                 00
                          0




                                  e
                       30




                               or
                              12

                              21

                              30

                              39

                              48

                              57

                              M
                                   Velocity(m/s)



  Figure 3.7 Comparison of juvenile and mature wood phase velocities in the radial direction.

       D. Conclusion
        When looking at the above figures, the phase velocities of the mature wood samples are
higher than those of the juvenile samples in the longitudinal direction. In both the radial and
tangential directions, the phase velocities tend to be fairly similar between the mature and
juvenile samples. In general, the longitudinal direction has the greatest velocities averaging 3700
m/s for juvenile and 4890 m/s for mature wood. Tangential and radial velocities tend to be
similar in value although lower than the longitudinal direction with the radial values at 2182 m/s
and 1983 m/s for juvenile and mature wood and tangential values of 1757 m/s and 1232 m/s
respectively.

        The differences in the phase velocities for the longitudinal versus radial and tangential
directions may be linked to the difference in microfibrillar angle between the juvenile and mature
wood. Figure 3.8 shows the microfibrillar orientation of cell wall layers. This difference in
orientation between the layers may affect the longitudinal direction differently than the tangential
or radial direction. The density of the juvenile wood is generally higher than that of the mature
wood and may contribute to the difference in ultrasonic wave velocities.




                                                  III-26
                                               Microfibril
                                               Orientation




Figure 3.8 Microfibrillar orientation in cell walls of wood.




                              III-27
IV.    Discussion of Available NDE
       A. Summary
        Eighteen timbers were evaluated using non-destructive techniques to determine the ratio
of juvenile and mature wood in the sample. The logs were struck by a hammer on one end, and
the elastic wave impulse was measured along the length of the log using a high speed data
acquisition system and two single axis pinducers. Data reduction was performed using fast
Fourier transforms to determine the input and output spectrums. Each log specimen was tested 5
times. Repetitive testing gave consistent spectrums; however, the bandwidth of the output signal
was within the range of 0 – 5 kHz regardless of the input signal.
       B. Procedure
        Logs were supported on two wood blocks to raise them off the floor and simulate free
boundary conditions on the external surface. Two single axis DC – 1.2 MHz (-3dB) model VP-
1093 (Valpey-Fisher, Hopkinton, MA) pinducers were mounted on 9/32” x           2 7/8” lag bolts
with silicon glue. The bolts were screwed into the log specimen along the length of the timber at
distances of 100 mm and 1250 mm from the impulse end of the log. Bolt were tightened until
only 1/8” of the bolt remained above the log surface. An impulse signal was generated by striking
the end of the timber sharply with a hammer. Each log was tested 5 times.
        Signal data was acquired using a digital storage oscilloscope (Tektronic TDS 520A,
Wilsonville, Oregon) and stored using a GPIB communication link between the oscilloscope and
a personal computer. Figure 4.1 shows the experimental set-up. The data acquisition was
performed at a rate of 250,000 samples/s for 2500 samples taking the input signal at 10% with no
delay. Matlab was then used to reduce and analyze the data using fast Fourier transforms. The
spectrums of the 5 hits for a sample were averaged, and the first eighty-four components were
used to run the analytical model. The analytical solution was then compared with the output data.




                                                 IV-28
               Digital Storage Oscilloscope TDS 520




                                                    CH 1 CH 2 AUX 1 AUX 2




             GPIB communication link
             To IEEE STD 488 Port (osc.)

                          Personal Computer




                                       Ultrasonic                            Ultrasonic
                                       PreAmp                                PreAmp



                               input                output           input                output


                    Pinducer                                                                 Pinducer
                    VP-1093                                                                  VP-1093



                                               Timber Sample



               Figure 4.1 Experimental set-up for data acquisition on log samples.


       C. Results
        Figures 4.2 and 4.3 show the average input and output spectrums for log 15C. Dotted
lines on the graph represent error bars of +/- 2 standard deviations. Figures 4.4 and 4.5 show
similar data for log 22C.




                                                             IV-29
           Log 15C - Avg. Input Spectrum +/- 2 SD

0.05
0.04
0.03
0.02
0.01
   0
-0.01 0        1            2            3            4          5   6

                                 Frequency (kHz)


               Figure 4.2 Average input spectrum for log 15C.




            Log 15C - Output Spectrum +/- 2 SD

 0.01
0.008
0.006
0.004
0.002
       0
-0.002 0        1            2            3           4          5   6

                                 Frequency (kHz)


               Figure 4.3 Average output spectrum for log 15C.




                                      IV-30
          Log 22C - Avg. Input Spectrum +/- 2 SD

0.04
0.03
0.02
0.01
   0
-0.01 0       1            2            3           4           5   6

                               Frequency (kHz)

               Figure 4.4 Average input spectrum for log 22C.




          Log 22C - Avg. Output Spectrum +/- 2 SD

0.08
0.06
0.04
0.02
   0
-0.02 0        1           2            3            4          5   6

                               Frequency (kHz)

              Figure 4.5 Average output spectrum for log 22C.




                                      IV-31
D. Conclusions
        The error bars on the graph denote +/- 2 standard deviations from the averaged
spectrums. The above figures show that repetitive testing of a particular log gives a
consistent spectrum. However, the output bandwidth is 0-5 kHz regardless of the input
signal. Looking at the output graphs for log 15C and log 22C, both signals disappear at
the 5 kHz mark even though the input signals are different.

       The results of the non-destructive evaluation of the timber samples suggests that
the hammer strike does not supply a signal that will be effective for determining the ratio
of juvenile and mature woods in timber samples. A higher frequency is needed for an
impulse input to use elastic waves as a tool for estimating small diameter timber
composition.




                                          IV-32
V.   Appendix
     A. Appendix: Overview of Project

                               Small Diameter Timber
                                      Project

                                    Log Beaking


              defects                 moisture           % juvenile wood


        defects don't matter                              Rich's Model


             assume %                                  run signals w/different
           juvenile wood                                    % of juvenile
              matters                                     & mature wood
                                                          (all frequencies)


          prove in best                                        predict
         possible manner                                    ~20 - 50 kHz
                                                              optimum


          combine 2 sets                                need good material
         of data & write up                                properties for
                                                         juvenile & mature
                                                           wood as input


                                                            re-run model


                                                             can find %
                                                           of mature and
                                                           juvenile wood


                                                         does % correlate
                                                         with log strength?


                                                       Experimental NDE



                                                  hammer                  ultrasonics
                                                  0-2 kHz               125 kHz -1 MHz

                                                            gap exists


                                                              invent
                                                              excitor


                                                       experimental NDE



                                                  33
       B. Appendix: Elastic Tensor Recovery

                1. Background
         Ultrasonic waves and signal analysis can be used to determine the full elastic
tensor of solid woods and wood based composites. Wood is considered to be an
orthotropic material with nine independent stiffness elastic tensor elements corresponding
to a stiffness matrix with six diagonal terms (C11, C22, C33, C44, C55, C66) and three off-
diagonal terms (C12, C13, C23) from which elastic engineering parameters can easily be
determined.

        An orthotropic material is characterized by six elastic parameters: three from the
modulus of elasticity or Young’s modulus (E1, E2, E3) and three from the modulus of
rigidity or shear modulus (G12, G13, G23). Ultrasonic waves can be used to test an
unknown specimen using longitudinal and shear waves in three directions: longitudinal,
tangential, and radial. These six signals can then be used to calculate the elasticity tensor
for the wood sample.

                2. General formulation of the problem
       The propagation of ultrasonic waves in an anisotropic material is given by
Christoffel’s equation:

                                  (Cijkl   V 2 ij ) Pm  0                               (1)

       where C ijkl is the stiffness elasticity tensor of the material,  is the density, P is
the polarization vector, V is the phase velocity of ultrasonic wave in the specimen,  ij is
the Kronecker symbol, and n i are the components of the unit vector n in the wave
propagation direction. Equation (1) supplies the relationship between the elasticity tensor
and the phase velocity of ultrasonic wave propagation.

        Hooke’s law relates all components of stress to strain in the material. Referring to
a fixed rectangular coordinate system ( x1, x2, x3 ), let  ij and  kl be the stress and strain
in a general anisotropic material. The stress-strain law as well as the strain-stress law can
be expressed as
                                         ij  C ijkl  kl                                   (2)

                                          kl  S ijkl  ij                                 (3)

       where S ijkl is the compliance elasticity tensor corresponding to the inverse of the
stiffness matrix  C . For an orthotropic material, the nine stiffness elasticity tensor
components ( C ijkl ) are independent. The same holds true for the compliance tensor ( S ijkl ).
Alternatively, equations (2) and (3) can be expressed as the matrix equation




                                                      34
                        1   C11                C12        C13              0              0     0   1 
                          C                    C 22       C 23             0              0     0   2 
                        2   21                                                                       
                         3  C 31               C 32       C 33             0              0     0   3 
                                                                                                                                (4)
                        23   0                  0           0             C 44            0     0  23 
                                                                                                       
                       13   0                   0           0              0          C 55      0  13 
                                                                                                      
                       12   0                   0           0              0              0    C66  12 
                                                                                                       

                             1   S 11           S 12       S 13            0              0    0  1 
                              S                S 22        S 23            0              0    0   2 
                            2   21                                                                   
                             3   S 31           S 32       S 33            0              0    0   3 
                                                                                                                                (5)
                            23   0                  0       0             S 44            0    0  23 
                                                                                                       
                            13   0                  0       0              0          S 55     0 13 
                                                                                                      
                            12   0                  0       0              0              0   S 66 12 
                                                                                                       

                                                             C    S  1                                                          (6)
Longitudinal and shear waves propagate along the three geometrical axes of symmetry of
the material: longitudinal, tangential and radial directions. The six stiffness elasticity
tensor elements ( C11 ,C 22 ,C 33 ,C 44 ,C 55 ,C66 ) can be obtained from equation (1) .

For a longitudinal wave:
                                   C11  V11 2  ,              longitudinal direction

                                    C 22  V22 2  , tangential direction

                                            C 33  V33 2  ,          radial direction                                                (7)

For a shear wave:
                                    C 44  V 44 2  , longitudinal direction

                                        C 55  V 55 2  , tangential direction

                                             C 66  V66 2  , radial direction                                                        (8)


The three off-diagonal elements ( C12 ,C13 ,C 23 ) can be calculated when a wave is
propagated in a direction outside of the three axes of symmetry for the material. The
equations are outlined as follows:
                                  2               2
                                                               
      C12  (n1 n2 ) 1 C11 n1  C 66 n2   V 2 C 66 n1  C 22 n2   V 2              2              2           1/ 2
                                                                                                                             C 66
      C13  (n1 n3 ) 1   C n
                             11 1
                                    2
                                         C55 n3
                                                    2
                                                          V C n  C n   V 
                                                                      2
                                                                                55 1
                                                                                          2
                                                                                                  33 3
                                                                                                         2        2 1/ 2
                                                                                                                             C55     (9)
      C 23  (n2 n3 ) 1   C n
                              22    2
                                        2
                                             C 44 n3
                                                        2
                                                           V C n  C n   V 
                                                                          2
                                                                                     44   2
                                                                                              2
                                                                                                   33 3
                                                                                                             2     2 1/ 2
                                                                                                                              C 44


                                                                              35
where V depends on the angle of the wave propagation through the sample.

         The three Young’s moduli ( E1 , E2 , E3 ) give the ratio of normal stress to normal
strain in the normal direction while the three shear moduli ( G12 ,G13 ,G23 ) give the ratio of
shear stress to shear strain in the orthotropic plane. They are defined by the following
equations:
                                             11                 22                33
                                                   E1 ,              E2 ,              E 33
                                              11                22                33

                                    12                       13                        23
                                         G12 ,                    G1 3 ,                    G23                  (10)
                                    12                       12                        23

       A comparison of Equation (10) with Equations (4), (5), and (6) leads to the
following relations:

                                      1                          1                             1
                              E1         ,            E2           ,               E3 
                                     S 11                       S 22                          S 33

                                                        1                             1                       1
                                              G 23         ,            G13             ,          G12 
                                                       S 44                          S 55                    S 66

                                                (11)

          For the wood samples in this paper, testing has only been completed using
longitudinal waves; hence, only three of the stiffness elasticity tensor elements
( C11 ,C 22 ,C 33 ) have been calculated.

                 3. Cross-correlation between signals of aluminum and wood cube
        Cross-correlation processing is used to determine the relative time delay
( ) between two waveforms in an experimental system. For example, the cross-
correlation between an ultrasonic signal in aluminum x( t ) and in wood y( t ) is a
sequence rxy ( t ) which can be defined as
                                                                             N
                                                              rxy (  )      x( t ) y( t   )
                                                                              N

                                                (12)
         where  is the time shift parameter and the subscript xy indicates the signals
being correlated. The sequence x( t ) is unshifted, and y( t ) is shifted by  units in time
to the right for  positive and to the left for  negative.

       In the most general case, the delay or dispersion is a function of the wavelength or
frequency. The cross-correlation rxy ( t ) provides a measure of similarity between the



                                                        36
power of the two signals so that the time delay corresponds to a delay between the peak
powers.

        In this case, the relative time delay ( ) for a narrow band signal is used to
estimate the phase velocity ( V11 , V22 , V33 ) of the ultrasonic waves propagating through
the wood. From these phase velocities, the stiffness elasticity tensor elements ( C11 , C 22 ,
C33 ) can be obtained as shown by the following equations:
                              V11  (d  Val ) /(d al  ( 11  Val )

                              V22  (d  Val ) /(d al  ( 22  Val )

                                     V33  (d  Val ) /(d  ( 33  Val )                         (13)

       where Val  6320 (m / s ) is the velocity of ultrasonic wave propagation in the
reference specimen (aluminum), d and d al are the distances the wave propagates through
the aluminum and wood cube samples, and  is the density of wood.

                4. Truncating the signal
If a signal with a long time window is used, an interference pattern is generated and
results in a received signal pattern similar to that of a standing wave. The phase velocities
obtained from the cross-correlation are not accurate unless the signal is windowed. To
ensure that multiple reflections are not included in the time signal, the time at which the
window is applied is determined by an adaptive calculation of the point when the wave
has traveled two complete passes through the specimen. To make an initial estimate of
flight time, a phase velocity is initially estimated based on the reference velocity. For all
experimental calculations, the initial estimates used are as followed: ( m / s )

           V11 ' = 3800      V 22 ' =1700        V 33 ' =2300           (for the juvenile wood)

           V11 ' = 5000      V22' = 1400          V 33 ' = 2300              (for the mature wood)
(14)


        Truncating the signal using a simple rectangular window results in more energy
appearing in the higher frequencies. Instead, a trapezoidal window is used in this
particular application as a compromise between simplicity and minimization of high
energy. Specifically, let the window function w( t ) be defined as followed:


                                                    1       t  ( 0.8 p )
                                       
                              w( t )  ( p  t ) /( 1  0.8 )       ( 0.8  t  p )
                                                              (t  p )
                                                      0
(15)



                                                    37
        Here p is flight time for the wave to travel two complete passes through the
specimen.
                5. Fourier transform of signals
       In order to analyze the energy distribution of the signals within the frequency
domain for both the reference specimen (aluminum) and testing specimen (wood),
Fourier transform should be performed. Figures 1 - 3 presented below are taken from one
juvenile wood cube tested ultrasonically using a longitudinal wave in the three directions:
longitudinal, tangential, and radial. Differences in the use of a rectangular versus
trapezoidal window are shown.


                           Figure 1.1 Longitudinal signal truncated using rectangular window
                 1
         Voltage
         (V)     0

                -1
                     0        0.5      1          1.5      2       2.5    3        3.5          4     4.5           5
                                                                Time(s)                                            -5
                                                                                                            x 10
                            Figure 1.2 Longitudinal signal truncated using trapezoidal window
                 1
         Voltage
         (v)     0

                -1
                     0        0.5      1      1.5          2      2.5     3        3.5          4     4.5          5
                                                               Time(s)                                         -5
                                                                                                            x 10
                          Figure 1.3 Comparison of Fourier transform between the signals above

               0.05                                                                  Rectangular window
         FFT
                                                                                     Trapezoidal window

                 0
                      0        0.005       0.01         0.015    0.02      0.025         0.03       0.035      0.04
                                                            Frequency(1/s)
               Figure 1. Comparison of Fourier transform between longitudinal
                 signals truncated using rectangular and trapezoidal window




                                                               38
                       Figure 2.1 Tangential signal truncated using rectangular window
       0.2
Voltage
(v)     0

      -0.2
             0        0.5      1          1.5       2       2.5    3       3.5          4      4.5           5
                                                         Time(s)                                            -5
                                                                                                     x 10
                     Figure 2.2 Tangential signal truncated using trapezoidal window
       0.2
Voltage
(v)     0

      -0.2
             0        0.5      1          1.5       2       2.5    3       3.5          4      4.5           5
                                                         Time(s)                                            -5
                                                                                                     x 10
                 Figure 2.3 Comparison of Fourier transform between the signals above
      0.04
FFT                                                                         Rectangular window
      0.02                                                                  Trapezoidal window

        0
             0        0.005        0.01         0.015    0.02      0.025         0.03       0.035       0.04
                                                    Frequency(1/s)
             Figure 2. Comparison of Fourier transform of tangential
             signals truncated using rectangular and trapezoidal window




                                                        39
                             Figure 3.1 Radial signal truncated using rectangular window
                0.2
         Voltage
         (v)     0

               -0.2
                      0      0.5     1          1.5       2       2.5    3       3.5          4      4.5           5
                                                               Time(s)                                            -5
                                                                                                           x 10
                              Figure 3.2 Radial signal truncated using trapezoidal window
                0.2
         Voltage
                 0
         (v)
               -0.2
                      0      0.5     1          1.5       2       2.5    3       3.5          4      4.5           5
                                                               Time(s)                                            -5
                                                                                                           x 10
                          Figure 3.3 Comparison of Fourier transform between signals above
               0.02
         FFT                                                                      Rectangular window
               0.01                                                               Trapezoidal window

                 0
                      0      0.005       0.01         0.015    0.02      0.025         0.03       0.035       0.04
                                                          Frequency(1/s)

                 Figure 3. Comparison of Fourier transform of radial signals
                      truncated using rectangular and trapezoidal window




        The above figures demonstrate that there is more energy in the higher frequencies
for signals truncated using a rectangular rather than trapezoidal window.




                                                              40
              6. Fourier transform of ultrasonic signal from reference specimen
              (aluminum) and testing specimen (juvenile wood)

Figures 4-6 show the Fourier transform of the aluminum reference specimen and a
juvenile wood test specimen truncated using a trapezoidal window. Transforms are
shown for all three directions: longitudinal, tangential, and radial.




         Figure 4.1 Ultrasonic signal of reference specimen (aluminum) truncated using trapezoidal window
                 5
       Voltage
       (v)       0

                -5
                     0     0.5        1       1.5      2       2.5     3      3.5         4   4.5           5
                                                            Time(s)                                        -5
                                                                                                    x 10

       Figure 4.2 Ultrasonic signal of testing specimen (wood) truncated using trapezoidal window
                 1
       Voltage
       (v)       0

                -1
                     0     0.5         1      1.5      2       2.5     3      3.5         4   4.5            5
                                                            Time(s)                                         -5
                                                                                                     x 10
                                 Figure 4.3 Fourier transforms of the two signals above
             0.05
       FFT                                                                            Reference
                                                                                      Juvenile wood

                 0
                     0    0.01       0.02    0.03    0.04   0.05    0.06     0.07    0.08     0.09          0.1
                                                       Frequency(1/s)
                         Figure 4. Fourier transform of signals for aluminum
                                  and wood in longitudinal direction




                                                           41
Figure 5.1 Ultrasonic signal of reference specimen (aluminum) truncated using trapezoidal window
         5
Voltage
(v)      0

        -5
             0     0.5        1      1.5      2       2.5     3      3.5          4   4.5           5
                                                   Time(s)                                         -5
                                                                                            x 10
Figure 5.2 Ultrasonic signal of testing specimen (wood) truncated using trapezoidal window
        0.2
Voltage
(v)       0

      -0.2
             0     0.5        1      1.5      2       2.5     3      3.5          4   4.5           5
                                                   Time(s)                                         -5
                                                                                            x 10
                         Figure 5.3 Fourier transforms of the two signals above
      0.05
                                                                              Reference
FFT                                                                           Juvenile wood

          0
              0    0.01      0.02    0.03    0.04   0.05    0.06     0.07     0.08    0.09         0.1
                                               Frequency(1/s)
                  Figure 5. Fourier transform for signals of aluminum
                           and wood in tangential direction




                                              42
         Figure 6.1 Ultrasonic signal of reference specimen (aluminum) truncated using trapezoidal window
                   5
         Voltage
         (v)       0

                 -5
                      0    0.5        1      1.5      2       2.5        3    3.5         4   4.5           5
                                                           Time(s)                                         -5
                                                                                                    x 10

             Figure 6.2 Ultrasonic signal of testing specimen (wood) truncated using trapezoidal window
                  0.2
         Voltage
         (v)        0

                -0.2
                       0   0.5        1       1.5      2          2.5    3    3.5         4   4.5           5
                                                               Time(s)                                     -5
                                                                                                    x 10
                                 Figure 6.3 Fourier transforms of the two signals above
                0.05
          FFT                                                                         Reference
                                                                                      Juvenile wood

                   0
                       0   0.01      0.02    0.03    0.04   0.05    0.06     0.07     0.08    0.09         0.1
                                                       Frequency(1/s)
                       Figure 6. Fourier transform of signals for aluminum and
                                        wood in radial direction




               7. Calculation of C11 ,C 22 ,C 33
       Signal processing shown above was used in conjunction with equation (7) to
determine C11 ,C 22 ,C 33 for the juvenile and mature wood specimens described in Section
II, Material Properties. Resulting phase velocities and their elastic constants are
displayed in the Results section.




                                                          43

				
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