Several Nondestructive 

Evaluation Techniques 

for Assessing Stiffness 

and MOE of SmallDiameter Logs 

Xiping Wang 

Robert J. Ross 

James A. Mattson 

John R. Erickson 

John W. Forsman 

Earl A. Geske 

Michael A. Wehr 

United States

Department of 

Agriculture 

Forest Service 

Forest 

Products 

Laboratory 

Research 

Paper 

FPL−RP−600 
Abstract 

Many forests of the United States have large areas that contain trees 

of small diameter, mixed species, and undefined quality. Because 

these areas are at risk from attack by insects, disease, and uncontrollable wild fires, it is essential to find ways to increase the incentive to harvest this small-diameter material. One way to do this is to 

develop cost-effective products for the fiber from these trees. 

Nondestructive technology needs to be developed to evaluate the 

potential quality of stems and logs obtained from trees in such 

ecosystems. Static bending, transverse vibration, and longitudinal 

stress wave techniques are frequently used to assess the modulus of 

elasticity (MOE) of lumber. Excellent correlations between MOE 

values obtained from these techniques have been reported. The 

objective of this study was to investigate the use of these techniques to evaluate the flexural stiffness and MOE of small-diameter 

logs. A total of 159 red pine and jack pine logs were obtained from 

northern Michigan and were assessed nondestructively using these 

techniques. Statistical comparisons between stiffness and MOE 

values obtained from each technique were then examined. Results 

of this study demonstrated that strong relationships exist between 

the log properties determined by the three techniques, longitudinal 

stress wave, transverse vibration, and static bending. Developed 

models allow for the prediction of static bending properties of logs 

at levels of accuracy previously considered unattainable. This 

indicates that any of these techniques can be used to sort smalldiameter logs with reasonable accuracy. 

Keywords: small-diameter log, nondestructive evaluation, stress 

wave, transverse vibration, static bending, 

modulus of elasticity (MOE), flexural stiffness 

May 2001 

Wang, X.; Ross, R.J.; Mattson, J.A.; Erickson, J.R.; Forsman, J.W.; Geske, 

E.A.; Wehr, M.A. 2001. Several nondestructive evaluation techniques 

for assessing stiffness and MOE of small-diameter logs. Res. Pap. 

FPL-RP-600. Madison, WI: U.S. Department of Agriculture, Forest Service, 

Forest Products Laboratory. 12 p. 

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Contents 

 Page 

Introduction .......................................................................... 1

Static Bending ................................................................... 1

Transverse Vibration......................................................... 2

Stress Wave....................................................................... 3

Materials and Methods ......................................................... 3

Results and Discussion ......................................................... 5

Physical Characteristics of Logs....................................... 5

Modulus of Elasticity of Logs .......................................... 6

Modulus of Elasticity Relationships ................................. 6

Flexural Stiffness Relationships ..................................... 10

Conclusions ........................................................................ 11

Literature Cited................................................................... 12
Several Nondestructive Evaluation 

Techniques for Assessing Stiffness 

and MOE of Small-Diameter Logs 

Xiping Wang, Research Scientist

Robert J. Ross, Supervisory Research General Engineer

Forest Products Laboratory, Madison, Wisconsin 

James A. Mattson, Project Leader

North Central Forest Experiment Station, Houghton, Michigan 

John R. Erickson, Research Scientist

John W. Forsman, Assistant Research Scientist

Michigan Technological University, Houghton, Michigan 

Earl A. Geske, Supervisory Electronics Technician (retired)

Forest Products Laboratory, Madison, Wisconsin 

Michael A. Wehr, Engineer

North Central Forest Experiment Station, Houghton, Michigan 

Introduction 

Many decades of inappropriate management practices, or 

lack of management altogether, have resulted in large 

amounts of U.S. forests being overstocked with smalldiameter trees of mixed species. These stands are typically 

low in value, and the value of the harvestable material will 

not cover the costs of needed management treatments. A 

specific example is the interior west region of the United 

States, where 39 million acres of ponderosa-pine-type forest 

have lost ecological integrity due to major changes in vegetative structure and composition. These changes have been 

caused by control of fire in an ecosystem where historically 

there were frequent, low-intensity stand maintenance fires. 

Exclusion of fire led to the current condition, and these 

stands are now at high risk of attack by insects, disease, and 

stand-destroying wildfires. Restoration, either mechanical or 

through prescribed fires, can cost $150 to $500 per acre. It is 

essential to find cost-effective products that can be produced 

from the materials available in these stands so that needed 

management operations such as thinning can be implemented to improve the condition of these stands. Economical 

and value-added uses for this material can help offset forest 

management costs, provide economic opportunities for many 

small, forest-based communities, and avoid future loss 

caused by catastrophic wildfires. The variability and lack of 

predictability of the strength and stiffness of this material 

cause problems in engineering applications. Cost-effective 

technology needs to be developed for evaluating the potential quality of stems and logs obtained from trees in such 

ecosystems. 

In the forest products industry, a variety of nondestructive 

evaluation (NDE) techniques are now being used to assess 

the engineering properties of structural lumber. Static bending, transverse vibration, and longitudinal stress wave techniques are frequently used to assess the modulus of elasticity 

(MOE) of lumber. Commercially available equipment that 

applies these techniques is readily available. The objective 

of this study was to investigate the use of these three 

techniques to evaluate the stiffness and MOE of smalldiameter logs. 

Static Bending 

Measuring the MOE of a member by static bending is the 

foundation of a commonly used process to grade structural 

lumber (Ross and Pellerin 1994). As currently employed for 

grading lumber, this relatively simple measurement involves 

utilizing the load–deflection relationship of a simply supported beam with different loading patterns (Fig. 1). The 

MOE is computed directly by using equations derived from 

the fundamental mechanics of materials.
2 

Figure 1a shows a standard bending configuration. The load 

is applied at the midspan of the beam. An ordinary dial 

gauge or electronic recording equipment is normally used to 

obtain load–deflection data. Bending MOE is then calculated 

from data taken from the linear elastic region of the load–

deflection curve: 

I

PL

δ = 48

MOE

3

 (1) 

where MOE is static modulus of elasticity (lb/in2

 (Pa)), P is 

load within the proportional limit (lb (N)), L is beam span 

(in. (m)), • is deflection at midspan within the proportional 

limit (in. (m)), and I is beam moment of inertia (in4

 (m4)). 

Figure 1b shows a general bending configuration used for 

testing structural lumber. The loads are equally applied at 

two points of the beam. Bending MOE is calculated by the 

following equation: 

I

Pa L a

δ

− = 48(3 4 ) MOE

2 2

 (2) 

where P is load (lb (N)), a is distance from the end support 

to nearest load point (in. (m)), L is beam span (in. (m)), • is 

midspan deflection (in. (m)), and I is beam moment of 

inertia (in4

 (m4)). 

Transverse Vibration 

Transverse vibration techniques have received considerable 

attention for NDE applications (Jayne 1959, Kaiserlik 1977, 

Pellerin 1965, Ross and others 1991, Ross and Pellerin 

1994). To illustrate these methods, an analogy can be drawn 

between the behavior of a vibrating beam and the vibration 

of a mass that is attached to a weightless spring and internal 

damping force (Fig. 2). In Figure 2, mass M is supported 

from a rigid body by a weightless spring whose elastic constant is denoted by K. Internal friction or damping is given 

by r, which represents the viscous damping coefficient of the 

dashpot. A forcing function equaling P0 sin •t or zero is 

applied for forced and free vibration, respectively. When M

is set into vibration, its equation of motion can be expressed 

by the following: 

Figure 1—Static bending configuration; (a) center-point bending, and 

(b) general bending (L, beam span; P, load within proportional limit; 

a, distance from end support to nearest load point). 

Figure 2—Mass-spring dashpot vibration model (left) and transverse vibrating beam 

(right) (P0 sin •t, a sinusoidal force; M, mass; K, elastic constant of the spring; 

r, viscous damping coefficient of the dashpot; L, beam span). 
3 

Kx P t

dt

dx r dtd x M  + = ω 

 



 +













 sin 0 2

2

(3)

Equation (3) can be solved for either K or r. 

A solution for K will lead to an expression for MOE where 

Ig

fr WL

12.65

MOE

2 3

= (4) 

for a beam freely supported at two nodal points and 

Ig

fr WL

2.46

MOE

2 3

= (5)

for a beam simply supported at its ends. 

In Equations (4) and (5), MOE is dynamic modulus of elasticity (lb/in2 or Pa), ƒr is resonant frequency (Hz), W is beam 

weight (lb or kg), L is beam span (in. or m), I is beam 

moment of inertia (in4

 or m4), and g is acceleration due to 

gravity (386 in/s2 or 9.8 m/s2). 

Stress Wave 

Several techniques that utilize stress wave propagation have 

been researched for use as NDE tools. Speed-of-sound 

transmission and attenuation of induced stress waves in a 

material are frequently used as NDE parameters (Ross and 

others 1994, 1996, 1999; Schad and others 1995; Wang 

1999; Wang and others 2000). 

To illustrate these techniques, consider application of onedimensional wave theory to the homogeneous viscoelastic 

bar (Fig. 3). After an impact hits the end of the bar, a compressive stress wave is generated in the bar that travels from 

left to right at a speed C. As the wave reaches the free end, it 

is reflected as a tension wave and begins traveling back 

down the bar. Energy is dissipated as the wave travels 

through the bar; therefore, although the speed of the wave 

remains constant, movement of particles diminishes with 

each successive passing of the wave. Eventually all particles 

of the bar come to rest. 

Monitoring the movement of a cross section near the end of 

such a bar in response to a propagating stress wave results in 

waveforms that consist of a series of equally spaced pulses 

whose magnitude decreases exponentially with time (Fig. 4). 

The propagation speed C of such a wave can be determined 

by coupling measurements of the time between pulses •t and 

the length of the bar L by 

t

L C

∆ = 2 (6)

The MOE can be computed using C and the mass density of 

the bar ρ: 

= ρ MOE 2 C (7) 

Although this equation was derived for an idealized, onedimensional case, it has been shown to exist for actual threedimensional members as long as the length of the wave is 

large relative to the member’s lateral dimensions. 

Materials and Methods 

A flow chart that outlines the experimental procedures is 

shown in Figure 5. First, a sample of small-diameter trees 

was selected from stands and harvested to obtain logs. 

Physical properties (diameter, moisture content, and green 

density) of logs were then measured. This was followed by a 

sequence of nondestructive tests using longitudinal stress 

wave, transverse vibration, and static bending techniques to 

obtain the MOE and flexural stiffness of these logs. Statistical analyses were then used to examine the relationships 

between log properties determined by the different 

techniques. 

In this study, 159 small-diameter logs (109 jack pine (Pinus 

banksiana Lamb.) and 50 red pine (Pinus resinosa Ait.)) 

were nondestructively evaluated. These logs came from trees 

that were grown on the Ottawa National Forest and the Lake 

Superior State Forest, both in northern Michigan. 

The jack pine logs used in this study were obtained from an 

overage stand of jack pine, which was beginning to show 

signs of deterioration. Ranger district personnel in the 

Ottawa National Forest were able to visually identify four 

categories of trees in this stand: live healthy trees (merchantable live), live trees that are showing signs of being under 

stress (suspect), trees that are dead but still containing 

Figure 3—Viscoelastic bar of length L subjected to an 

impact. 

Figure 4—Theoretical response of the end of a 

viscoelastic bar (cross section) in response to a 

propagating stress wave. 
4 

merchantable material (merchantable dead), and dead trees 

that have deteriorated to the point of having no merchantable 

material (unmerchantable dead). To manage these jack pine 

stands, forest managers have been holding commercial 

salvage sales on considerable acreages. To properly estimate 

the value of these stands, better information on the value of 

each of the four categories of trees is needed. Trees in each 

of these categories were selected for this study. The estimated ages of these jack pine trees ranged from 50 to 

70 years old. The diameter at breast height (DBH) of sampled trees ranged from 5.0 to 12.2 in. (127 to 310 mm). 

Red pine logs were obtained from 38-year-old research plots 

in the Lake Superior State Forest. The objective of the original research on these plots was to examine the growth of red 

pine with time at various stocking levels and correlate volume yield with financial yield at different initial stocking 

levels. Plots at five levels of stocking were available: 220, 

320, 420, 620, and 820 trees per acre. Ten trees were harvested from each of the stocking level plots. The DBH of 

sampled trees ranged from 4.7 to 11.5 in. (11.9 to 29.2 cm). 

After these sampled trees were harvested, a 16-ft- (4.9-m-) 

long butt log was bucked from each tree on site and then 

transported to the Forestry Sciences Laboratory, USDA 

Forest Service, North Central Forest Experiment Station, 

Houghton, Michigan, for various nondestructive testing. 

Upon arrival at the Forestry Sciences Laboratory, a 2-ft- 

(61-cm-) long section from each end of the butt log was cut 

off and sent to the USDA Forest Service, Forest Products 

Laboratory, in Madison, Wisconsin, for pulping studies. 

The remaining 12-ft- (3.7-m-) long logs were then used for 

this study. To determine moisture content (MC) of sampled 

trees, we cut three disks from each tree, one from the butt, 

the middle, and the top. Green weight and ovendry weight of 

these disks were then obtained and used to determine tree 

MC. For each 12-ft- (3.7-m-) long log, the green weight and 

the diameters of both ends were measured to obtain the 

green density and the moment of inertia of the log. 

Each log was first evaluated using a longitudinal stress wave 

technique to obtain an estimate of dynamic MOE (stress 

wave MOE (MOEsw)) of the log. Figure 6 shows the experimental setup for stress wave measurements on the logs. An 

accelerometer was attached to one end of the log. A stress 

wave was introduced to the log through a hammer impact on 

the opposite end, and the resulting stress wave was recorded 

with a personal computer. A detailed description of the 

instrumentation and analysis procedures can be found in a 

previous article (Ross and others 1994), and a discussion of 

its application to large specimens is included in Schad and 

others (1995). From stress wave measurement, the stress 

wave speed (C) in a log was determined by Equation (6). 

Dynamic MOE (MOEsw) was then calculated using 

Equation (7). 

After stress wave tests, the logs were vibrated using a transverse vibration technique. The experimental setup is shown 

in Figure 7. A digital oscilloscope and an accelerometer 

were used in this test. The log under test was supported at 

one end by a knife-edge support and at the opposite end by a 

point support. The accelerometer was located in the middle 

of the log and glued on the upside surface, where the bark 

was removed or polished to improve the contact between 

the accelerometer and the log. The log was then set into 

vibration by impacting the middle part of the log with a 

rubber hammer. The free vibration response of the log was 

observed in the oscilloscope. The signal observed was a 

series of pulses with a gradually decreasing (decaying) 

amplitude. The vibrational parameter measured was fundamental natural frequency. The dynamic MOE of logs determined by transverse vibration technique (vibration MOE 

(MOEv)) was calculated from Equation (5). 

Figure 5—Flowchart of study. 
5 

Static bending tests were then performed on the logs to 

obtain the static MOE (MOEs

) and flexural stiffness (EI) of 

these logs. Measuring MOE of a member by static bending 

techniques has been widely considered the foundation of 

lumber grading and NDE of wood and wood-based materials. However, this technique is rarely used to evaluate the 

MOE of logs as a standard method. Consequently, no standard testing procedure exists for testing small-diameter logs. 

However, we assumed the MOEs

 of logs to be the true MOE 

for logs, and we used it to evaluate the dynamic MOE of 

logs determined by stress wave and transverse vibration 

techniques. A Metriguard (Pullman, Washington) Model 312 

bending proof tester (Fig. 8a) was used to conduct static 

bending tests on all logs. Figure 8b shows the bending configuration (general bending) we used. The testing machine 

was originally designed for proof-loading dimensional lumber. To test logs, we modified the two end supports to fit the 

geometrical shape of small-diameter logs (Fig. 9). The modified supports allowed testing of logs with a maximum diameter of 12 in. (30.5 cm). The span between two supports 

was set at 115.5 in. (2.93 m). The distance from loading 

point to the nearest support was 38.5 in. (0.98 m), which was 

one-third of the span. A load was applied to the log through 

two bearing blocks. Deflection was measured in the central 

region, a zone of pure bending without shear deformation. 

The log under test was first preloaded to 100 lb (445 N), and 

the deflection was set to zero. This procedure was mainly 

used to improve the contact between the log, supporters, and 

bearing blocks and thus to eliminate the effect of bark on the 

deflection measurement. The log was then loaded to 0.2-in. 

(5.08-mm) deflection. The load value corresponding to this 

deflection was then recorded. Static MOE (MOEs

) of the log 

was calculated using Equation (2). 

Results and Discussion 

Physical Characteristics of Logs 

Table 1 summarizes the physical characteristics of the red 

pine and jack pine logs. The average diameters of the butt 

logs obtained from the trees ranged from 4.4 to 10.2 in. 

(11.2 to 25.9 cm) for red pine and from 4.7 to 11.0 in. 

(11.9 to 27.9 cm) for jack pine. This is a typical diameter 

range for small-diameter timber (Wolfe 2000). For both 

species, the average MC exceeded the fiber saturation point 

(about 30%). However, the red pine logs apparently had 

much higher MC level than the jack pine logs. The individual values ranged from 88% to 145% for red pine logs and 

from 31% to 65% for jack pine logs. The low MC level for 

jack pine logs was caused by the fact that they came from 

different categories that included live, suspect, and dead 

trees. The suspect and dead trees had already lost a lot of 

moisture by the time they were harvested. Therefore, the MC 

of some logs obtained from dead trees was close to or even 

lower than the fiber saturation point. 

Also, red pine logs have higher density than jack pine logs. 

The green density for the red pine logs ranged from 48.0 to 

56.5 lb/ft3

 (770 to 900 kg/m3), and that for jack pine logs 

ranged from 28.7 to 53.7 lb/ft3

 (460 to 860 kg/m3). The 

lower value and large range of density for jack pine logs was 

also due to the different tree categories. 

Figure 6—Experimental setup for stress wave test on 

logs. 

Figure 7—Experimental setup for transverse vibration 

test on logs. 

Figure 8—(a) Static bending test on logs, and 

(b) bending configuration (a, distance from end 

support to nearest load point; P, load within 

proportional limit; L, beam span). 
6 

Jack pine logs differ from red pine logs in cross section of 

stem shape and in straightness. Red pine logs are mostly 

round and very straight. Jack pine logs tend to have a more 

irregular shape in cross section and a more curved stem, 

which could introduce errors in the determination of density 

and moment of inertia of these logs. 

Modulus of Elasticity of Logs 

Results obtained from various NDE measurements of both 

red pine and jack pine logs are summarized in Table 2. The 

basic statistics of dynamic MOE (determined by stress wave 

and transverse vibration techniques) and static MOE (determined by static bending technique within elastic region) for 

both species are given in Table 2. 

The static MOE (MOEs

) of logs ranged from 0.45 to 

1.21 ×106

 lb/in2 (3.10 to 8.34 GPa) with a mean value 

of 0.80 ×106

 lb/in2 (5.52 GPa) for red pine and 0.17 to 

1.48 ×106

 lb/in2 (1.17 to 10.20 GPa) with a mean value of 

0.81 ×106

 lb/in2 (5.58 GPa) for jack pine. The stress wave 

technique produced a higher estimate of MOE for both 

species. For red pine logs, the mean MOEsw was 11.8% and 

18.8% greater than its vibrational and static counterpart, 

respectively. For jack pine logs, the mean MOEsw was 21.6% 

and 24.7% greater than its vibrational and static counterparts, respectively. We believe that the higher value of 

MOEsw could be related to the wave propagation mechanism, 

the dimension of the logs, or the moisture state of the wood 

in the logs. 

In previous studies (Wang 1999, Wang and others 2000), we 

found that the stress wave behaved differently in logs than in 

small, clear wood and lumber because of the relative large 

size of the logs. As the wave travels through a log in the 

longitudinal direction, the outer portion of the wood (mature 

wood) may have a dominating effect on the propagation of 

waves. This led to a higher stress wave speed for a log compared with small, clear specimens cut from the log, which 

increased the value of MOEsw and in turn overestimated the 

MOE of the log. It was also found that the diameter-tolength ratio could be a critical factor that may affect stress 

wave behavior in logs. Quantitative analyses of the overestimation in MOEsw of logs have not been reported. 

Compared with MOEsw of logs, the dynamic MOE of logs 

determined from transverse vibration technique (MOEv) is 

much closer to static MOE of logs. The MOEv of red pine 

logs ranged from 0.58 to 1.22 ×106

 lb/in2 (4.00 to 8.40 GPa), 

and the range for jack pine logs was 0.25 to 1.47 ×106

 lb/in2

(1.72 to 10.14 GPa). For both species, the mean value of 

MOEv was about 7% greater than the mean MOEs

. 

Modulus of Elasticity Relationships 

Statistical analysis procedures were used to examine the 

relationships between the various MOE of red pine and jack 

pine logs. The results obtained from regression analyses are 

presented in Tables 3 and 4.

Univariable Regression Models 

The correlations among various MOE could be represented 

by linear regression models (y = a + bx). The results of the 

comparison of the three different techniques are reported in 

terms of correlation coefficients. These results show how 

reliable the method might be for prediction purposes. The 

square of the correlation coefficient expresses the percentage 

of the total variability explained by the regression line. 

In general, the dynamic MOE (MOEsw and MOEv

) of logs 

was very closely correlated with the static MOE (MOEs

) for 

both red pine and jack pine logs. The correlation coefficients 

were 0.87 (MOEsw versus MOEs

) and 0.97 (MOEv versus 

MOEs

) for red pine logs. Those for jack pine logs were 0.77 

(MOEsw versus MOEs

) and 0.92 (MOEv versus MOEs). The 

linear regression analyses indicated that the developed regression models were statistically significant at the 0.01 

confidence level. 

Figure 9—Modified fixture of supports used for static 

bending test. 
7 

Table 1—Physical characteristics of red pine and jack pine logsa 

 

DBHb

 of trees 

 (in.) 

Diameter of butt 

logs (in.) 

Density 

(lb/ft3) 

Moisture content 

(%) 

 Species and log group 

Number 

of logs 

Average  Minimum– 

 maximum 

Average  Minimum– 

 maximum Average 

Minimum–

maximum 

Average  Minimum– 

 maximum 

Red pine 

220c 10 9.82 8.50–10.35 9.16 7.90–10.13 53.3 49.7–56.0 113.6 114.1–144.6 

320 10 9.87 8.12–11.10 9.16 7.63–10.16 53.8 52.3–56.7 121.1 114.5–140.4 

420 10 8.67 7.88–9.02 7.99 7.37–8.31 54.2 51.9–56.5 115.3 88.2–132.6 

620 10 7.23 5.48–8.70 6.79 5.30–8.16 50.9 48.0–52.8 109 121.7–134.5 

820 10 6.72 4.70–7.86 6.24 4.40–7.42 52.7 49.4–54.4 116.7 116.7–145.2 

Jack pine 

Merchantable live 30 9.4 7.4–11.0 8.59 6.83–9.90 47.04 42.18–53.73 65 50.9–107.2 

Suspect 29 9.4 6.4–12.0 8.47 6.14–10.99 42.26 29.01–51.38 54.1 40.2– 74.3 

Merchantable dead 32 7.9 5.0–12.2 7.11 4.67–10.56 34.91 28.66–43.34 36.8 21.0– 55.7 

Nonmerchantable dead 13 7.6 6.3–9.7 6.65 5.58–8.50 34.58 29.46–42.16 31.2 20.6– 38.4 

a

1 in. = 25.4 mm, 1 lb/ft3 = 16.01 kg/m3. b

DBH, diameter at breast height. 

c

Number of trees per acre. 

Table 2—Modulus of elasticity (MOE) of red pine and jack pine logsa 

Dynamic MOE (MOEsw) Dynamic MOE (MOEv

) Static MOE (MOEs) 

 

Species 

Number 

of logs Mean SD 

Minimum– 

maximum Mean SD 

Minimum– 

maximum Mean SD 

Minimum– 

maximum 

Red pine 50 0.95 0.110 0.76–1.22 0.85 0.164 0.58–1.22 0.80 0.187 0.45–1.21 

Jack pine 109 1.11 0.230 0.47–1.84 0.87 0.246 0.25–1.47 0.81 0.235 0.17–1.48 

a

MOEsw, dynamic MOE of a log determined by stress wave technique; MOEv, dynamic MOE of a log determined by transverse 

 vibration technique; MOEs, static MOE of a log determined by general static bending; SD, standard deviation. 

Table 3—Results of linear regression analyses of various moduli of elasticity (MOE) of red pine and jack pine logsa

Linear regression model y = a + bx

 Species y x a b r

2

r Syx F-testb

Red pine 

 MOEv MOEsw −0.3927 1.3101 0.77 0.88 0.079 162.1*** 

 MOEs MOEsw −0.5986 1.4740 0.75 0.87 0.095 143.1*** 

 MOEs

 MOEv −0.1443 1.1105 0.95 0.97 0.044 835*** 

Jack pine 

 MOEv MOEsw −0.0318 0.8167 0.58 0.76 0.160 142.4*** 

 MOEs MOEsw −0.0644 0.7883 0.60 0.77 0.150 150.9*** 

 MOEs

 MOEv 0.0425 0.8782 0.85 0.92 0.092 567.1*** 

Combined 

 MOEv

 MOEsw 0.0478 0.7748 0.55 0.74 0.150 182.9*** 

 MOEs

 MOEsw 0.0089 0.7555 0.53 0.73 0.151 172.4*** 

 MOEs MOEv 0.0820 0.9175 0.86 0.93 0.082 946.9*** a

MOEsw, MOE of a log determined by stress wave method; MOEv, MOE of a log determined by transverse vibration method; 

 MOEs, MOE of a log determined by general static bending; r

2

, coefficient of determination; r, correlation coefficient; 

Syx, standard error of estimate. b

 ***Highly significant (0.01 confidence level). 
8 

Figures 10 to 13 show the relationships of dynamic MOE 

predicted by stress wave technique to dynamic MOE predicted by transverse vibration technique and static bending 

MOE for both species. The red pine logs produced a better 

correlation (r = 0.87 to 0.88) than the jack pine logs 

(r = 0.76 to 0.77). This could be attributed to the geometrical 

differences between the two species. It was evident that the 

irregular shape (irregular cross section and curved stem) of 

some jack pine logs could introduce errors in diameter 

measurements, thus causing errors in density and MOE 

determination, especially in MOEsw determination. 

Also, the plotted data points were more heavily concentrated 

below the 45° line than above, thus indicating that the stress 

wave technique yields higher MOE values than its vibrational and static counterparts. As was discussed earlier, the 

higher value of MOEsw could have been caused by several 

factors such as wave propagation mechanism, log size, and 

moisture state of wood. Of these factors, log size (diameter 

D and length L) seems more important because it could 

affect stress wave behavior in logs. Wang (1999) reported 

that high diameter-to-length ratio D/L could cause significant changes in wave propagation path in the longitudinal 

direction in stress wave measurements of logs. Therefore, it 

seems that the effect of log size should not be neglected in 

the MOE regression models. 

The relationships between MOEv and MOEs

 of red pine and 

jack pine logs are shown in Figures 14 and 15. The MOE 

correlations from transverse vibration tests were significantly improved compared with those from stress wave tests. 

In the relationship between MOEv and MOEs

, it was found 

that the two species could be combined and represented as a 

single population. The correlation coefficient relating the 

MOEv to MOEs

 was 0.93 for the two species combined. This 

value clearly indicates that the variation caused by these two 

species does not affect the relationship. 

Multivariable Regression Models 

For MOEsw, the univariable linear regression models resulted 

in a correlation coefficient of 0.77 to 0.87 with static bending MOE. These values, although significant, indicate a 

Figure 10—Relationship of stress wave MOE (MOEsw) to 

static MOE (MOEs) for red pine logs (1 lb/in2 = 6.9 kPa). 

Figure 11—Relationship of stress wave MOE (MOEsw) to 

static MOE (MOEs) for jack pine logs (1 lb/in2 = 6.9 kPa). 

Table 4—Results of regression analyses relating static bending modulus of elasticity (MOEs) to stress wave MOE (MOEsw) 

and ratio of log diameter to log length (D/L)

a

Regression model y = ax1

b x2c

 Species y x1 x2 a b c r

2 r Syx

Red pine MOEs MOEsw D/L 0.0968 0.9078 −0.7326 0.91 0.953939 0.055 

Jack pine MOEs MOEsw D/L 1.7826 1.1957 −0.5060 0.74 0.860233 0.120 

a

MOEsw, MOE of a log determined by stress wave method; MOEs, MOE of a log determined by general 

 static bending; r

2

, coefficient of determination; r, correlation coefficient; Syx, standard error of estimate. 

0.6

0.8

1.0

1.2

1.4

1.6

MOEs of logs (x106 lb/in2

) 0.4 0.6 0.8 1.0 1.2 1.4 1.6

MOEsw of logs (x106 lb/in2)

Red pine 

y = –0.5986 + 1.4740x

r

2 = 0.75 

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

MOEs of logs (x106 lb/in2

)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

MOEsw of logs (x106

 lb/in2)

Jack pine 

y = –0.0644 + 0.7883x

r

2 = 0.60 
9 

Figure 12—Relationship of stress wave MOE (MOEsw) 

to static MOE (MOEs) for red pine logs 

(1 lb/in2

 = 6.9 kPa). 

Figure 13—Relationship of stress wave MOE (MOEsw) 

to vibration MOE (MOEv

) for jack pine logs 

(1 lb/in2 = 6.9 kPa). 

greater scatter of points about the regression line than had 

occurred in MOEv. In an effort to obtain a better prediction 

model for MOE of logs, a multivariable regression model 

relating MOEs

 to MOEsw and diameter-to-length ratio was 

developed. The mathematical regression models used in this 

analysis were assumed to be of the following form: 

b c y ax x 1 2 = (8) 

where y is modulus of elasticity being estimated; a, b, c are 

empirical constants; x1 is nondestructive parameter MOEsw; 

and x2

 is ratio of log diameter to log length. 

The MOE of logs predicted by Equation (8) was then compared with the static bending MOE of logs. Results of the 

Figure 14—Relationship of vibration MOE (MOEv) 

to static MOE (MOEs) for red pine logs 

(1 lb/in2

 = 6.9 kPa). 

Figure 15—Relationship of vibration MOE (MOEv

) 

to static MOE (MOEs) for jack pine logs 

(1 lb/in2

 = 6.9 kPa). 

regression analyses and values for the constants in the equations are summarized in Table 4. 

The relationship between stress-wave-predicted MOE using 

the developed multivariable model and the static MOE of 

logs is shown in Figures 16 and 17. Figures 16 and 17 indicate that a strong relationship existed between stress-wavepredicted MOE and static MOE. Compared with the univariable linear regression model, the multivariable models 

showed significantly better correlation. The correlation 

coefficient r increased from 0.87 (red pine) and 0.77 (jack 

pine) for the univariable model to 0.95 (red pine) and 0.86 

(jack pine) for the multivariable model. This showed that the 

diameter-to-length ratio had an interactive effect that contributed significantly when used in conjunction with MOEsw. 

0.6

0.8

1.0

1.2

1.4

1.6

MOEv of logs (x106 lb/in2

) 0.4 0.6 0.8 1.0 1.2 1.4 1.6

MOEsw of logs (x106 lb/in2)

Red pine 

y = –0.3927 + 1.3101x

r

2 = 0.77 

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

MOEsw of logs (x106 lb/in2)

Jack pine 

y = –0.0318 + 0.8167x

r

2 = 0.58 

MOEv of logs (x106 lb/in2

)

0.6

0.8

1.0

1.2

1.4

1.6

MOEs of logs (x106 lb/in2

) 0.4 0.6 0.8 1.0 1.2 1.4 1.6

MOEv of logs (x106

 lb/in2)

Red pine 

y = –0.1443 + 1.1105x

r

2 = 0.95 

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

MOEs of logs (x106 lb/in2

)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

MOEv of logs (x106

 lb/in2)

Jack pine 

y = 0.0425 + 0.8782x

r

2 = 0.85 
10 

Figure 16—Relationship of stress-wave-predicted MOE 

to static MOE (MOEs

) for red pine logs (1 lb/in2 = 6.9 kPa). 

Figure 17—Relationship of stress-wave-predicted MOE to 

static MOE (MOEs) for jack pine logs (1 lb/in2 = 6.9 kPa). 

Flexural Stiffness Relationships 

Of the parameters that can be measured nondestructively 

(density, appearance, MOE, and stiffness), stiffness is used 

most frequently to predict the strength of wood materials. 

Therefore, it is important to know the relationships between 

the stiffness determined by the three techniques covered in 

this report. 

Flexural stiffness is expressed as the product of the moment 

of inertia (I) and MOE in bending. For logs, the moment of 

inertia is given by 

64

D4

I π = (9)

where D is the average diameter of a log (in. (m)). 

When the MOE of logs determined by these techniques is 

known, the flexural stiffness of logs can be easily calculated. 

The relationships between various log stiffness (stress wave 

EI, vibration EI, and static EI) are shown in Table 5 and 

Figures 18 to 20. Red pine and jack pine logs showed no 

distinction in the stiffness relationships. Therefore, we combined these two species and treated them as a single population. 

The results revealed that the correlations between these 

nondestructively determined stiffness values were extraordinarily strong. In Figures 18 and 19, the static EI and vibration EI were plotted as a function of stress wave EI. Compared with the MOE relationships, the stress wave technique 

and the transverse vibration and static bending techniques 

were significantly better correlated in terms of flexural 

stiffness. Regression analyses indicated that a second-order 

polynomial regression model (y = a + bx + cx2

) could best 

Table 5—Results of regression analyses of flexural stiffness (EI) of logs of both species determined by different techniquesa

Regression model y = a + bx + cx2 or y = a + bx

y x a b c r

2 r Syx F-testb

EIs EIsw 11.6810 0.7530 −0.0003 0.94 0.97 23.37 2,246*** 

EIv EIsw 5.1330 0.9035 −0.0005 0.97 0.98 18.03 1,226*** 

EIs EIv −1.8262 0.9434 — 0.94 0.97 22.83 2,575*** 

a

EIsw, stiffness of log determined by longitudinal stress wave technique; EIv, stiffness of log 

 determined by transverse vibration technique; EIs

, stiffness of log determined by general 

 static bending; r

2

, coefficient of determination; r, correlation coefficient, Syx, standard error of estimate. b

***Highly significant (0.01 confidence level). 

0.6

0.8

1.0

1.2

1.4

1.6

MOEs of logs (x106 lb/in2

) 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Stress-wave-predicted MOE of logs (x106 lb/in2)

Red pine 

MOEs=0.0968 MOEsw

0.9078 (D/L)

-0.7326

r

2 = 0.91 

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

MOEs of logs (x106 lb/in2) 

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Stress-wave-predicted MOE of logs (x106 lb/in2)

Jack pine 

MOEs = 1.7826 MOEsw

1.1957 (D/L)-0.506

r

2 = 0.74
11 

Figure 18—Stress wave flexural stiffness (EI) versus 

static EI (1 lb·in2 = 0.00287 N·m). 

Figure 19—Stress wave flexural stiffness (EI) versus 

vibration EI (1 lb·in2 = 0.00287 N·m). 

Figure 20—Vibration flexural stiffness (EI) versus static 

EI (1 lb·in2 = 0.00287 N·m). 

fit the experimental data. The correlation coefficients were 

0.97 (stress wave EI versus static EI) and 0.98 (stress wave 

EI versus vibration EI), respectively. In other words, the 

developed regression models accounted for 97% and 94% of 

observed behavior. 

Figure 20 shows the relationship between flexural stiffness 

of logs measured by transverse vibration and static bending 

techniques. Just as in the case of the MOE relationship, a 

linear regression model was found to be the best fitting 

function to the experimental data. The correlation coefficient 

was 0.97, indicating that 94% of observed behavior has been 

accounted for. 

Conclusions 

The results of these experiments show that small-diameter 

red pine and jack pine logs can be successfully evaluated by 

longitudinal stress wave, transverse vibration, or static bending techniques. The dynamic MOE (MOEsw and MOEv) of 

logs was found to be well correlated with the static MOE for 

both species. 

The experimental results demonstrated that the stress wave 

technique was sensitive to the geometrical imperfections of 

logs. Round and straight logs produced better correlation 

between MOEsw and MOEs

 than did logs that were not round 

or straight. More importantly, the diameter-to-length ratio 

D/L had an interactive effect that contributed significantly 

when used in conjunction with MOEsw. The developed multivariable model relating MOEs

 to MOEsw and diameter-tolength ratio was found to be a better predictor for static 

MOE of logs. This could allow for the prediction of static 

bending properties of logs using stress wave techniques at 

levels of accuracy previously considered unattainable. 

The results from transverse vibration tests demonstrated that 

a significant improvement in predicting MOE of logs was 

achieved compared with the results from stress wave tests. 

In regard to the relationship of MOEv

 and MOEs, transverse 

vibrational parameters were found to be less sensitive to the 

geometrical imperfections of logs than were the stress wave 

parameters. Red pine and jack pine logs therefore could be 

combined and represented as a single population in the 

prediction model. 

Strong relationships were found between the various nondestructively determined log stiffnesses. Compared with the 

MOE relationships, the stress wave technique and the transverse vibration and static bending techniques were significantly better correlated in terms of flexural stiffness. 

100

200

300

400

500

600

700

800

900

Els of logs (x106 lb•in2

)

0 100 200 300 400 500 600 700 800 900

EIsw of logs (x106 lb•in2)

Jack pine Red pine

y = 11.6810 + 0.7530x - 0.0003x

2 

r

2 = 0.94 

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700 800 900

Jack pine Red pine

y = 5.1330 + 0.9035x - 0.0005x

2

r

2 = 0.97 

EIv of logs (x106 lb•in2

)

EIsw of logs (x106 lb•in2)

100

200

300

400

500

600

EIs of logs (x106 lb•in2

)

0 100 200 300 400 500 600

EIv of logs (x106 lb•in2)

Jack pine Red pine

y = –1.8262 + 0.9434x

r

2 = 0.94 
12 

Literature Cited 

Jayne, B.A. 1959. Vibrational properties of wood as indices 

of quality. Forest Products Journal. 9(11): 413–416. 

Kaiserlik, J.H.; Pellerin, R.F. 1977. Stress wave attenuation as an indicator of lumber strength. Forest Products 

Journal. 27(6): 39–43. 

Pellerin, R.F. 1965. A vibrational approach to nondestructive testing of structural lumber. Forest Products Journal. 

14(3): 93–101. 

Ross, R.J.; Pellerin, R.F. 1994. Nondestructive testing for 

assessing wood members in structures: A review. Gen. Tech. 

Rep. FPL–GTR–70 (Rev.). Madison, WI: U.S. Department 

of Agriculture, Forest Service, Forest Products Laboratory. 

40 p. 

Ross, R.J.; Geske, E.A.; Larson, G.L.; Murphy, J.F.

1991. Transverse vibration nondestructive testing using a 

personal computer. Res. Pap. FPL–RP–502. Madison, WI: 

U.S. Department of Agriculture, Forest Service, Forest 

Products Laboratory. 

Ross, R.J.; De Groot, R.C.; Nelson, W.J. 1994. Technique 

for nondestructive evaluation of biologically degraded wood. 

Experimental Techniques. 18(5): 29–32. 

Ross, R.J.; McDonald, K.A.; Green, D.W.; Schad, K.C.

1996. Relationship between log and lumber modulus of 

elasticity. Forest Products Journal. 47(2): 89–92. 

Ross, R.J.; Willits, S.W.; Segen, W.V; Black, T.; 

Brashaw, B.K.; Pellerin, R.F. 1999. A stress wave based 

approach to NDE of logs for assessing potential veneer 

quality. Part 1. Small-diameter ponderosa pine. Forest 

Products Journal. 49(11/12): 60–62. 

Schad, K.C.; Kretschmann, D.E.; McDonald, K.A.; 

Ross, R.J.; Green, D.W. 1995. Stress wave techniques for 

determining quality of dimensional lumber from switch ties. 

FPL–RN–0265. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 

Wang, X. 1999. Stress wave-based nondestructive evaluation (NDE) methods for wood quality of standing trees. 

Houghton, MI: Michigan Technological University. 

Ph.D. dissertation. 

Wang, X.; Ross, R.J.; Erickson, J.R.; Forsman, J.W.; 

McGinnis, G.D.; De Groot, R.C. 2000. Nondestructive 

methods of evaluating quality of wood in preservativetreated piles. Res. Note FPL–RN–0274. Madison, WI: 

U.S. Department of Agriculture, Forest Service, 

Forest Products Laboratory. 9 p. 

Wolfe, R. 2000. Research challenges for structural use of 

small-diameter round timbers. Forest Products Journal. 

50(2): 21–29.