Transverse Vibrations 

of Wood-Based 

Products: Equations 

and Considerations

Joseph F. Murphy

United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Note

FPL–RN–0324
January 2011

Murphy, Joseph F. 2011. Transverse vibrations of wood-based products: 

equations and considerations. Research Note FPL-RN-0324. Madison, WI: 

U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 4 p.

A limited number of free copies of this publication are available to the 

public from the Forest Products Laboratory, One Gifford Pinchot Drive, 

Madison, WI 53726–2398. This publication is also available online at 

www.fpl.fs.fed.us. Laboratory publications are sent to hundreds of libraries 

in the United States and elsewhere.

The Forest Products Laboratory is maintained in cooperation with the 

University of Wisconsin. 

The use of trade or firm names in this publication is for reader information 

and does not imply endorsement by the United States Department of 

Agriculture (USDA) of any product or service.

The USDA prohibits discrimination in all its programs and activities on the 

basis of race, color, national origin, age, disability, and where applicable, 

sex, marital status, familial status, parental status, religion, sexual orientation, genetic information, political beliefs, reprisal, or because all or a part 

of an individual’s income is derived from any public assistance program. 

(Not all prohibited bases apply to all programs.) Persons with disabilities 

who require alternative means for communication of program information (Braille, large print, audiotape, etc.) should contact USDA’s TARGET 

Center at (202) 720–2600 (voice and TDD). To file a complaint of discrimination, write to USDA, Director, Office of Civil Rights, 1400 Independence 

Avenue, S.W., Washington, D.C. 20250–9410, or call (800) 795–3272 

(voice) or (202) 720–6382 (TDD). USDA is an equal opportunity provider 

and employer.

Abstract

Four equations are presented to determine bending stiffness 

using transverse vibration. These equations are used for 

constant cross-section products, panels, rectangular crosssection products, and logs with and without taper. Practical 

considerations for their use are discussed and concluding 

remarks are included. 

Keywords: transverse vibration, bending stiffness, constant 

cross section, panels, rectangular cross section, logs, taper

Content

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

Analytical Solutions............................................................1

 Equations.........................................................................1

 Constant Cross-Section Products....................................1

 Panels..............................................................................1

 Rectangular Cross-Section Products...............................2

 Logs.................................................................................2

Practical Considerations......................................................2

 Transverse Vibrations of Logs........................................2

 Dynamic and Static Modulus of Elasticity .....................2

 Log Tests.........................................................................3

 Lumber Tests...................................................................3

 Effect of Moisture Content .............................................3

Concluding Remarks...........................................................4

References...........................................................................4
Transverse Vibrations of Wood-Based 

Products: Equations and Considerations 

Joseph F. Murphy, Research Engineer 

Forest Products Laboratory, Madison, Wisconsin 

Introduction 

For the past 15 plus years, the Forest Products Laboratory, 

specifically the Engineering Mechanics and Remote Sensing 

Laboratory, has been using transverse vibration of woodbased products to nondestructively measure bending stiffness, which then can be correlated with strength (Green and 

others 2006). We use the basic vibration equation to measure bending stiffness that was determined by Timoshenko 

and others 1974) and can be found in general engineering 

books, such as that authored by Young (1989). Whereas the 

basic equation was determined for slender isotropic materials with constant cross section, we have successfully applied 

it to wood, wood-based composites, and even tapered logs. 

This paper presents the equations to measure bending stiffness, analytical solutions for specific classes of products, 

and practical considerations. 

Analytical Solutions 

Equations 

Four equations are used to determine bending stiffness using transverse vibration for various types of wood products. 

The most important equation is for products with a constant 

cross section. The bending stiffness EI is calculated from 

two terms. The first term uses just the fundamental frequency and the length of the span on which the specimen is 

simply supported. The second term is the specimen mass 

per unit length. No information about the cross section 

is required, as it is buried in the moment of inertia. 

Equation (1) accounts for overhang as noted in Murphy 

(1997) and is an excellent approximation for a span/specimen length (S/L) ratio ≥ 0.80. The other three equations use 

the same two right-hand terms of Equation (1) but multiply 

these terms by geometric modifiers. 

Constant cross-section products

 (1)

Panels

(2)

Rectangular cross-section products

(3)

Logs

(4a)

where

(4b)

b is horizontal width of specimen, 

cb butt circumference of log, 

ct tip circumference of log, 

E modulus of elasticity, 

EI bending stiffness, 

Eo modulus of elasticity of log, 

f fundamental frequency of vibration, 

g gravitational (acceleration) constant 

(980.665 cm/s2, 386.089 in/s2), 

h vertical height of specimen, 

I moment of inertia, 

L length of specimen, 

S support span (< L), 

W total weight of specimen, and 

p 3.1415926... 

Constant Cross-Section Products 

Equation (1) is valid for all wood and wood–plastic members with constant cross section. We use it for “I” and “T” 

beams as well as glulam members. The plane of vibration 

is vertical with the member on edge. The moment of inertia 

does not need to be known because displacement is important for engineering applications, and bending stiffness (EI) 

is used directly without separating out modulus of elasticity. 

Panels 

Equation (2) is used for panel products. The panel must be 

vibrated in one dimension so that it vibrates like a very wide 

beam. Care must be taken so that the panel does not vibrate 

in two directions simultaneously. Here again bending stiffness (EI) per unit width is important to determine displacement, and specific knowledge of moment of inertia is not 

needed for plywood or oriented strandboard (OSB). 

EI = f 

2S 4 W

π2

/4 gL

f 

2S 4 W 1

EI /b = 

f 

2S 4 W 12

π2 E = /4 gL bh3

f 

2S 4 W 64π3 1

π2

/4 gL cb θ Eo = 4

θ =

π2

/4 gL b

(m2 + 6mn + n2) (1 + a + a2)

32 3

a = ct

 /cb

m = 2 + (a – 1)(1 + S/L) 

n = 2 + (a – 1)(1 – S/L) 
Research Note FPL–RN–0324

2

Rectangular Cross-Section Products 

Equation (3) is used for dimension lumber as well as any 

solid sawn (or wood–plastic) member that has a rectangular cross section. For wood–plastic or even glulam beams, 

modulus of elasticity is an effective modulus because uniform material properties across the gross cross section are 

assumed. In some wood–plastic composites, the modulus 

of elasticity for the skin and core differs, but an effective 

modulus of elasticity can be determined for a particular 

cross section. 

Logs 

Equation (4) is used for logs, either doweled (θ = 1) or 

tapered. When the tip circumference is equal to the butt 

circumference, θ = 1. The second term on the right-hand 

side of the θ = ... equation, Equation (4b), accounts for the 

volume of a tapered cylinder compared with a straight cylinder. The first term on the left accounts for measuring the tip 

and butt circumferences at the ends of the log and not at the 

supports and the effect of taper on the fundamental root of 

the frequency equation. While this term is technically most 

accurate, if the ratio of tip circumference to butt circumference is greater than or equal to 0.80, then, as an excellent 

approximation, the log can be treated as a straight cylinder 

(θ = 1) with a constant circumference equal to the average 

of the tip and butt circumferences (Murphy 2000b, fig. 1). 

Practical Considerations 

Transverse Vibrations of Logs 

Practical considerations for the transverse vibration of logs 

require the use of two load cells because one end of a log 

can weigh up to twice as much as the other end. Figure 2 

shows a log end support. When swung to one side, the disengaging lever raises the support off the load cell. The 120° 

angled ridges provide a corner edge for centering the log. 

With the log centered, the load cell measures half the weight 

at the end of the log. The total weight of the log is calculated 

by multiplying each load cell reading by 2 and adding. If the 

log is bowed, it is placed bowed side down for safety and to 

ensure only one plane of vibration, i.e., the vertical plane. A 

sewing tape measure is used to measure circumference, assuming the log is round in cross section. 

If care is not taken while debarking, debarked logs may 

have processing taper at both ends. In this case, measuring 

the circumference at each end underestimates the volume, 

resulting in unrealistically high modulus of elasticity values 

(Eo). To avoid this problem, we measure the circumference 

“z” distance from the ends, with c1 near the tip end and c2

near the butt end. We then use the following equations to 

estimate the tip and butt circumferences. 

where

Consider now the transverse vibration of a log with the bark 

still attached. The bark can act like springs or nonrigid supports, which affect the fundamental frequency of the system 

(Murphy 2000a.) Moreover, taking circumference measurements around the bark without accounting for the bark 

thickness (i.e., without subtracting 2π × thicknessbark from 

the circumference) skews the resulting log modulus of elasticity Eo. Each variable—log volume, weight, density, the 

moment of inertia, and weight/unit length—includes both 

log and bark. Therefore, the resulting Eo is off by the value 

of I, while the fundamental frequency compensates for the 

(log + bark weight)/unit length. 

Dynamic and Static Modulus of Elasticity 

We have found that using transverse vibration gives essentially the same modulus of elasticity in bending as a 

dead load static bending test with the specimen in the same 

orientation. Modulus of elasticity values are different when 

the orientation is changed. For example, a piece of dimension lumber (nominal 2- by 4-in., standard 38- by 89-mm; 

hereafter called 2 by 4) is transversely vibrated flatwise. The 

flatwise modulus of elasticity is the same as that from a flatwise dead load static test. When the specimen is subjected 

to an edgewise dead load static test, the edgewise modulus 

of elasticity can be 15% less than the flatwise modulus of 

elasticity. The difference is not dynamic as opposed to static 

testing, but flatwise as opposed to edgewise orientation. 

This misconception is apparently the result of dynamically 

testing dimension lumber flatwise to estimate a static edgewise modulus of elasticity. Dynamic and static modulus of 

elasticity values are the same in the same orientation. Following are two examples of dynamic and static testing with 

the specimens in the same orientation. 

Figure 1. Transformed fundamental root of the frequency 

equation as a function of the degree of taper. cS, circumference of log at the support near the butt; co, circumference 

of log at the support near the tip.

ct = c2 [2 +(β – 1)(1 +1/X )]/2

cb = c2 [2 +(β – 1)(1 –1/X )]/2

X = 1 – (2z/L)

c1 ≤ c2

β = c1/c2
Transverse Vibrations of Wood-Based Products: Equations and Considerations 

3

Log Tests 

In the first example, 119 doweled logs were transversely 

vibrated (Green and others 2006). Then, without moving 

the logs on the supports, we added a 50-lb (22.7-kg) dead 

weight at midspan and measured midspan deflection. We 

then plotted dynamic versus static modulus of elasticity 

(Fig. 3). 

Lumber Tests 

In the second example, we tested 59 pieces of 2 by 4 lumber, varying in length from 96 to 102 in. (244 to 259 cm). 

For 50 pieces, the span was 96 in. (244 cm); for the remaining 9 pieces, the span was 92 in. (234 cm). Each piece was 

weighed. After the transverse vibration test (and without 

moving the lumber from the support stands) we added a 5-lb 

(2.3 kg) dead weight, P, at midspan and measured midspan 

deflection, δ. Cross section was not measured. We plotted 

calculated EI/S3 from the vibration tests using Equation (1), 

against calculated EI/S3 from the static dead weight tests 

using δ = PS3/(48EI) (Fig. 4). A best-fit line yielded a slope 

of 1.00977 and an offset of 0.009947 with a correlation 

coefficient r2 of 0.997. The offset corresponds to a difference of only 0.0036 in. (0.0914 mm) in the static deflection 

measurement. With twisted boards, we found that adding the 

5-lb (2.3-kg) weight could increase the twist and skew the 

deflection measurement if care was not taken. 

With the spans used and the moment of inertia of a 

flatwise 2 by 4, EI/S3 = 1.0 corresponds approximately 

to E = 1.0 × 106 lb/in2 (6.9 GPa). 

Effect of Moisture Content 

If the wood is not frozen, then the bending stiffness EI as 

measured is at the moisture content at the time of measurement. The added weight of the water affects the frequency 

of vibration appropriately such that the product frequency 

and weight (f 2W) remains a constant. This can be proven 

Figure 2. Log end support: (a) unassembled; (b) assembled. Load cell and disengaging lever shown. 

Figure 3. Dynamic versus static modulus of elasticity for 

119, dry, 9-in. (229-mm) uniform-diameter logs. 

Figure 4. Dynamic stiffness versus static stiffness. 
4

Research Note FPL–RN–0324

by measuring the EI of a member and then attaching a chain 

along the length of the member and measuring EI again. 

The added weight of the chain increases the weight per unit 

length, W/L, but the frequency of vibration, f, is slowed such 

that f 2W does not change. 

Concluding Remarks 

The Engineering Mechanics and Remote Sensing Laboratory of the Forest Products Laboratory uses four equations 

in the nondestructive assessment of bending stiffness properties of structural wood and wood–plastic members. The 

first two terms of all the equations account for overhang of 

the test specimen over the supports and are very accurate for 

a ratio of support span to specimen length (S/L) equal to or 

greater than 0.80. The result of Equation (1) is used in the 

other equations. The general Equation (1) is merely modified by geometric specifics to apply transverse vibration to 

panels (Eq. (2)), rectangular cross sections (Eq. (3)), and 

logs (Eq. (4)). If the log taper is not severe (ct/cb ≥ 0.80), 

then the log can be treated accurately as a straight cylinder 

(θ = 1) with a constant circumference equal to the average 

of the tip and butt circumferences. 

Dynamic bending stiffness EI as measured by transverse 

vibration is the same as static bending stiffness EI as measured by dead load if the support setup and specimen orientation are not changed. 

References

Green, D.W.; Gorman, T.; Evans, J.W.; and Murphy, J.F. 

2006. Mechanical grading of round timber beams. Journal 

of Materials in Civil Engineering. 18(1):1–10. 

Murphy, J.F. 1997. Transverse vibration of a simply supported beam with symmetric overhang of arbitrary length. 

Journal of Testing and Evaluation. 25(5):522–524. 

Murphy, J.F. 2000. Commentary on factors affecting transverse vibration using an idealized theoretical equation. Res. 

Note FPL–RN–0276. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 4 p. 

Murphy, J.F. 2000. Transverse vibration of a simply supported frustum of a right circular cone. Journal of Testing 

and Evaluation. 28(5):415–419. 

Timoshenko, S.; Young, D.H.; and Weaver, W. Jr. 1974. 

Vibration problems in engineering. 4th ed. New York: John 

Wiley & Sons. 

Young, W.C. 1989. Roark’s formulas for stress and strain. 

6th ed. New York: McGraw-Hill, Inc.