United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Paper

FPL 468

Mechanical

Property Ratios

A Measure of Flake Alignment

Robert L. Geimer
Abstract Contents

An equation of the form MOE = eμ SGα MOE:R±(0.5) is useful

from a theoretical standpoint in predicting modulus of elasticity

(MOE) of oriented flakeboards from the board variables of

specific gravity (SG) and flake alignment, where alignment is

defined by the ratio of bending stiffness properties (MOE:R). In

practice, MOE values are predicted with greater accuracy if the

exponential value of MOE:R is allowed to vary between flake

types.

The above equation may be used to predict modulus of rupture

(MOR) values only if μ and α are redefined and the modulus of

rupture ratios (MOR:R) are used. MOR:R values for

homogeneous aligned boards are less than MOE:R values at the

same degree of flake alignment (measured as an average flake

angle on the surface of the board). Both MOE:R and MOR:R

vary between flake types at the same degree of angular flake

orientation.

Keywords: Alignment, predictive formulae, mechanical property,

flakeboard.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Discussion ...................................

Predicting MOE .............................

Predicting MOR ...............................

Summary ....................................

Literature Cited ................................

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Page

1

1

2

2

4

7

7

8

Derivation of equation (MOE Pa) (MOE Pe) =

(MOE Random)2

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Derivation of equation MOE Pe = e

u

SGα

MOE:R( δ –1)

Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Derivation of

equation δ =

 ln MOE Pa – ln MOE Random

ln MOE:R

Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Derivation of (MOE Pa)(1–δ ) (MOE Pe)δ

= MOE Random

Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Examples of Adjustment of Test Data for Specific

Gravity and Flake Alignment

March 1986

Geimer, Robert L. Mechanical property ratios–a measure of flake alignment.

Res. Pap. 468. Madison, WI: U.S. Department of Agriculture, Forest Service,

Forest Products Laboratory; 1986. 10 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 53705-

2398. Laboratory publications are sent to over 1,000 libraries in the United States

and elsewhere.

The Laboratory is maintained in cooperation with the University of Wisconsin.

9

9

10
Mechanical

Property Ratios

A Measure of Flake Alignment

Robert L. Geimer

Forest Products Laboratory, Madison, WI

Introduction Objective

Since 1975 the commercial production of structural flakeboard

has increased over 400 percent. Most of the new mills

manufacture or have the capability of manufacturing oriented

strandboard (OSB), a flakeboard characterized by flake

alignment. The design of oriented strandboard, like that of other

engineered products, is governed by the known or predicted

properties of its components. The performance of this composite

panel is determined by the distinctive composition of its layers.

The purpose of this report is to expand on equations that predict

certain board properties and to present formats that can be used

by both researchers and manufacturers.

In 1975 research was undertaken at the Forest Products

Laboratory (FPL), Madison, WI, to establish basic relationships

between flakeboard properties and the manufacturing variables of

flake alignment and board specific gravity (SG). The mechanics

of flake alignment (Geimer 1976) and the effect of flake

alignment and board SG on the board properties of bending,

compression, and tension (Geimer 1979); internal bond and shear

(Geimer 1981); and dimensional stability (Geimer 1982) have

been described. The effect of mixing flake types (Geimer 1980)

has also been reported. 

 1
Discussion

In the equation

Strength or stiffness = e

μ

 SGαSVR±β

(1)

Predicting MOE

Theoretical Considerations

where

SG

SVR

= specific gravity

= sonic velocity ratio (a measure of flake alignment)

sonic velocity parallel

= sonic velocity perpendicular

μ, α, β = constants for a particular flake type and/or type of

where μ, α, and β are constants for the MOE property, based on

flake type and flake size.

loading; i.e. bending, tension, compression, shear The converse of this

the manufacturing variables of flake alignment and board SG are

related to the board properties of bending, tension, compression,

and interlaminar shear with reasonable accuracy (Geimer 1976,

1981). To some extent, this relationship is applicable to the

dimensional stability property of linear expansion (Geimer 1982).

An important feature of the equation’s format is that test data

from the parallel (Pa) to alignment direction (+β) and the

perpendicular (Pe) to alignment direction (−β) can be combined

to calculate the regression equation.

The use of sonic velocity (SV) measurements has several

advantages. Sonic velocity ratios as opposed to static stiffness or

strength ratios measure alignment independent of the predicted

physical property. Regression coefficients developed using SVR

data show a high degree of correlation to the data. In addition,

the equipment used to measure sonic velocity is relatively

inexpensive and easy to use, and has potential for “on line”

measurement systems. Despite these advantages, many

laboratories and manufacturing sites do not have the capability to

measure sonic velocity and use the ratio of bending properties as

the standard indicator of alignment. In the following discussion,

equations will be developed that incorporate the modulus of

elasticity ratio (MOE:R) and the modulus of rupture ratio

(MOR:R) as a means to predict MOE and MOR.

SVR = MOE:R2 β

—

1

(3)

may be substituted into equation (1) creating a new format useful

to those who measure alignment with MOE:R

MOE = e

μ

 SGα MOE:R±(0.5) (4)

Since the MOE:R for a board with random flake distribution is 1,

its MOE is defined simply as

MOE Random = e

μ

 SGα

(5)

Equation (4) can then be presented in two forms:

MOE Aligned = MOE:R±(0.5)

MOE Random

or (Appendix A)

(MOE Pa) (MOE Pe) = (MOE Random)2

Both forms can be used to ascertain the degree of alignment

needed for a desired increase in design stiffness (Geimer 1980).

Note that when MOE:R is used to quantify the alignment effect,

and the MOE:R and SVR are proportional on a log scale, the

exponent is always 1⁄2; i.e. the alignment term is invariant for all

flake types. In contrast, when SVR is used to describe alignment,

its exponent, β, can vary with the flake type.

Dividing equation (1) defining bending modulus of elasticity in

the Pa direction by the equivalent equation for MOE in the Pe

direction provides an expression for the MOE:R

MOE Pa e

μ

 SGαSVR+ β

 = SVR MOE:R = 2 β

MOE Pe

 =

e

μ

 SGα

SVR − β (2)

(6)

(7)

The above equations can also be used to predict MOR values if μ

and a are redefined for MOR and alignment is defined in terms

of MOR:R.

MOE for boards of varying SG made with 0.020- by 3-inch

Douglas-fir disk flakes is plotted as a function of MOE:R in

figure 1 using constants μ = 13.73 and α = 1.29 (table 1; Geimer

1979).* The use of log-log paper simplifies the graphics as the

slope of the line is always 0.5. The vertical position of the lines

describing boards of different SG will change with flake types,

dependent on the values of μ and α.

*Note: μ values given in table 1 provide MOE values of lb/in.2 × 10–3. To obtain actual

values add 6.91 to the value of μ (where 6.91 = ln 1000).

2
In actual practice μ and α values are unknown but may be

computed if boards of the same material are constructed at two

different SG levels. Using equation (5) to describe MOE for both

SG levels and solving the two equations simultaneously for α

gives:

(α = ln

MOE Random1

MOE Random2

SG1

(8)

ln

SG2

MOE Random values are either measured directly on boards

having random flake distribution or are calculated from aligned

boards using a modified form of equation (6)

MOE Random = MOE Pa

MOE:R0.5 (9)

Substituting the value of α in equation (5), μ is calculated as

μ = ln MOE Random1 – α ln SG1 (10)

Practical Applications

Many factors, including the density gradient, sample-to-sample

variations, and inadequate spread in test-board target SGs, lessen

the accuracy of predicting equation constants and MOE values. I

have found that MOE values will follow a more predictable

pattern of response to SG and alignment if μ is established at

some fixed value, and if both a and the exponential value of

MOE:R (0.5 in eq. [4]) are treated as variables.

The bending MOE of a board is then defined as

MOE Pa = eμ

 SGα MOE:Rδ (11)

for a board tested in the aligned direction and

MOE Pe = e

μ

 SGα MOE:R( δ–1) (12)

for a board tested perpendicular to the aligned direction

(Appendix B).

If μ is fixed, α can be easily calculated from random board data

using a modified form of equation (10)

α =

 (ln MOE Random) – μ

ln SG (13)

Data from an aligned board is needed to compute δ as follows

(Appendix C);

δ =

 ln MOE Pa – ln MOE Random

ln MOE:R (14)

Note that MOE Random must be referenced to the same SG as

that of the aligned board. However, this MOE value may be

calculated from a board of a different SG using equation (5) when

α and μ have been established as described above.

Equations (1 I) and (12) can be converted to a form similar to

equation (7) (Appendix D).

(MOE Pa)( 1−δ) (MOE Pe)δ = MOE Random (15)

An example of using equations (11 through 14) to adjust test

boards to a common SG or flake alignment is given in

Appendix E. The curve describing the relationship between MOE

and MOE:R is depicted in figure 1. Unlike the curves derived in

equation (4) where δ is fixed at 0.5, the slope of the MOE Pa

curve is different from the slope of the MOE Pe curve. Note that

when δ is allowed to vary, the slope of the curves can also vary

between flake types. When adjusting test values, I have found

that δ can vary between 0.36 and 0.48 (Geimer 1980;

unpublished data on flake classification).

The actual percent of flake alignment (i.e., the average relative

orientation of individual flakes) necessary to achieve a given

board property ratio (SVR, MOE:R, etc.) can also vary between

flake type.

Equations of the form

percent alignment = X ln MOE:R (16)

have been developed by measuring the alignment angles of flakes

on the surface of oriented flakeboard. Percent of alignment is

computed as 4 5 45 – θ where θ is the average of angles which the

flakes make with the cardinal direction of alignment. Values of X

given for several flake types have been published (Geimer 1981).

Disregarding two extreme data sets, the values of X ranged from

36.4 to 29.8, indicating that maximum MOE:R obtainable at

100 percent alignment varies between 15.6 and 28.7. The

relationship between MOE:R and percent alignment shown on the

X axis in figure 1 is based on a pooled set of data from several

flake types (Geimer 1981). This regression equation

percent alignment = 30.7 ln MOE:R (17)

limits the maximum attainable MOE:R to 25.98. 

 3
Predicting MOR The equation

As mentioned previously, bending strength can be predicted using

equation (4) provided μ and α are redefined and alignment is

measured in terms of the MOR:R. Bending MOR does not

respond to either SG or alignment to the same degree as bending

MOE. I have found that the MOR:R for homogeneous aligned

boards is always less than MOE:R at the same degree of average

angular flake orientation. This observation has been documented

by others (Hunt 1983; Snodgrass et al. 1973). However, Kajita

(1983) showed MOR:R of three-layer boards to exceed MOE:R.

The boards were constructed using randomly dispersed ring flakes

for the core and aligned disk flakes for the faces.

β 2 ln MOR:R

β 1

=

ln MOE:R

The relationship between the bending MOR:R and MOE:R,

derived by dividing equation (3) by the equivalent equation for

MOR, is defined as

β 2

is derived from equation (18) by taking the natural log of both

sides and transposing terms. Using values β1 = 1.23 and

β2 = 1.00 (table 1) for 0.020-inch-thick by 3-inch long

Douglas-fir flakes, the MOR:R is compared to the MOE:R on the

X axis in figure 1. Whereas MOE:R reaches 25.98 at 100 percent

alignment, the maximum MOR:R obtainable is only 14.13. The

dotted line in figure 1 represents the values of MOR at various

levels of alignment for a board with a SG of 0.65. μ2 and α2

values taken from table 1 (Geimer 1979) are 8.9 and 1.6,

respectively. Substituting the equivalent of MOR:R from

equation (18) into the MOR version of equation (4) provides a

convenient means of estimating MOR from MOE:R values:

MOR:R = MOE:Rβ1

(18)

where β1 and β2 are the exponents in equation (1). relating to

MOE and MOR respectively. While the actual β exponents are

difficult to obtain, the ratio β1

β2

can easily be computed from

measured MOE and MOR test data.

While all of the preceding examples and equations were directed

at or developed for bending properties, the same methods can be

used to predict MOE and strength values for compression,

tension, and interlaminar shear.

(19)

MOR = e

μ2 SGα2 MOE:R

±

 β2

2β1

(20)

4
Figure 1.—MOE and MOR. Plotted as a function of the modulus of elasticity ratio (MOE:R) where 

 5
Table 1 .—Constants for equation

Stiffness or strength = e

μ

 (specific gravity)a

Tension Compression

Thickness

Flake typea

Length Machine

Exponentb

Bending

MOEc MORd

MOEc

Stress at

maximum

load

MOEc

Stress at

maximum

load

In. (mm)

0.020 3

(.508) (76)

.020

(.508)

.020 .75

(.508) (19)

.010 3

(.254) (76)

.020 3

(.508) (76)

.045

(1.143)

In. (mm)

1.5

(38)

2

(51)

Disk μ

α

β

R

2

Disk μ

α

β

R

2

Ring μ

α

β

R

2

Disk μ

α

β

R

2

Disk μ

α

β

R

2

Ring μ

α

β

R

2

DOUGLAS-FIR

6.82 8.9

1.29 1.6

1.23 1.00

97.6 98.0

6.95 8.9

1.29 1.7

1.25 .90

97.6 95.0

6.77 8.7

1.48 2.0

1.12 .69

96.9 96.0

OAK

6.90 8.33 6.79 8.43

1.59 1.59 1.39 2.06

1.31 1.14 1.27 .75

94.8 96.2 95.3 92.7

7.06 8.43 6.91 8.42

1.57 1.74 1.34 1.94

1.32 1.00 1.25 .76

96.6 95.5 94.6 94.8

7.03 8.23 6.77 8.43

1.45 2.09 1.49 2.15

1.27 .84 1.25 .57

83.0 94.3 90.0 95.2

6.98 9.1

1.59 2.4

1.25 1.07

97.1 92.0

6.83 9.0

1.20 1.5

1.28 1.2

98.4 96.0

723 8.63 6.92 8.44

1.55 2.07 1.49 3.08

1.27 1.31 1.26 .40

93.6 94.5 88.5 71.4

7.08 8.44 6.72 8.52

1.55 1.78 1.63 2.55

1.32 1.42 1.22 .60

93.8 95.3 93.1 87.8

6.83 8.9 7.13 8.33 6.95 8.63

2.06 2.6 2.04 2.36 2.46 2.92

1.07 .18 1.00 .92 1.14 .52

97.3 95.0 93.1 96.2 95.0 98.2

aAll flakes were random width.

bR2

 is coefficient of determination.

cMOE is modulus of elasticity. - Note: μ values given in table 1 provide MOE values of lb/in.2

 × 10–3 To obtain actual values add 6.91 to the value of μ (where

6.91 = ln 1000).

dMOR is modulus of rupture.

6
Summary Literature Cited

The relationship between the flakeboard properties of either

bending, compression, tension, or interlaminar shear, and the

manufacturing variables of board SG and flake alignment can be

established by fabricating just a few test boards. The equation

that relates these properties

MOE = eμ

 SGα MOE:R±(0.5)

(4)

uses the MOE:R as a measure of flake alignment. Theoretically,

the exponent, 0.5, is applicable to all flake types.

Geimer, R. L. Flake alignment in particleboard as affected by

machine variables and particle geometry. Res. Pap. FPL 275.

Madison, WI: U.S. Department of Agriculture, Forest Service,

Forest Products Laboratory; 1976.

Geimer, R. L. Data basic to the engineering design of

reconstituted flakeboard. In: Proceedings, 13th annual

particleboard symposium. Pullman, WA: Washington State

University; 1979.

In practice, better correlation can usually be obtained if the

exponent of the MOE:R is treated as a variable. The value of δ in

the equation that defines the MOE tested in the

parallel-to-alignment direction

Geimer, R. L. Predicting flakeboard properties: Improvement in

bending by alining a mixture of flakes. In: Proceedings, 14th

annual particleboard symposium. Pullman, WA: Washington

State University; 1980.

MOE Pa = e

μ

 SGα MOE:Rδ

(11)

Geimer, R. L. Predicting shear and internal bond properties of

flakeboard. Holz als Roh-und Werkstoff. 39: 409-415; 1981.

Geimer, R. L. Dimensional stability of flakeboards as affected

by board specific gravity and flake alignment. Forest Products

Journal. 32(8): 45-52; 1982.

has been found to range from 0.36 to 0.48 with various flake

types. The MOE:R corresponding to a particular measured degree

of angular flake orientation has also been found to vary with flake

type. Theoretical maximum MOE:Rs obtainable with different

flakes vary from 15.6 to 28.7.

Hunt, M.; Hoover, W. Design and development of composite

structural panels with mixed hardwoods. Unpublished progress

report. Cooperative research agreement FP-81-0351. West

Lafayette, IN: Purdue University, Department of Forestry and

Natural Resources, Madison, WI: U.S. Department of

Agriculture, Forest Service, Forest Products Laboratory; 1983

April l-June 30.

The above equations can be used to predict MOR values only if Kajita, Hiromu. Oriented particleboards from thinnings of

μ and α are redefined and the MOR:R is used as a measure of Japanese Cedar. Presented at IUFRO all-division 5 conference,

alignment. MOR does not respond to alignment in the same Madison, WI; 1983 June 27-July 5.

manner as MOE. Experimental data show that MOR:R of Snodgrass, J. D.; Saunders, R. J.; Syska, A. D. Particleboard

homogeneous aligned boards is less than MOE:R values at the of aligned wood strands. In: Proceedings, 7th Washington State

same degree of angular flake orientation. University symposium on particleboard; 1973: 415-418. 

7
Appendix A

Derivation of equation (MOE Pa) Derivation of equation MOE Pe = e

μ

SGa

(MOE Pe) = (MOE Random)2 MOE:R(d – 1)

Appendix B

Given equation (6) for MOE Pa:

MOE Pa

MOE Random

Square both sides

MOE Pa 2 MOE Pa

= MOE Random MOE Pe

transpose terms

(MOE Pa)2

MOE Pa MOE Pe = (MOE Random)2

simplify

(MOE Pa) (MOE Pe) = (MOE Random)2

Similarly, given equation (6) for MOE Pe:

MOE Pe (0.5)

MOE Random =

MOE Pe

MOE Pa

square both sides

MOE Pe 2 MOE Pe

= MOE Random MOE Pa

transpose terms

(MOE Pe)2

MOE Pe MOE Pa = (MOE Random)2

simplify

(MOE Pe) (MOE Pa) = (MOE Random)2

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

Given:

MOE Pa = e

μ

 SGα MOE:Rδ

(B1)

Since

MOE Pa MOE:R = MOE Pe (B2)

substitute in B1 MOE Pa = eμ

 SGa MOE Pa

 MOE Pe

 δ

(B3)

multiply equation (B3) by MOE Pe

MOE Pa

MOE Pa

MOE Pe MOE Pa = e

μ

 SGα

MOE Pe

MOE Pa δ MOE Pe

MOE Pa (B4)

reduce equation and invert factor MOE Pe

MOE Pa

MOE Pe = e

μ

 SGα MOE Pa

MOE Pe

MOE Pa

MOE Pe

δ (–1)

(B5)

combine terms

MOE Pe = eμ SGα MOE:R(δ – 1) (B6)

8
Appendix C

Derivation of equation

d = ln MOE Pa – ln MOE Random

ln MOE:R

from equation (11)

MOE Pa = e

μ

 SGα MOE:Rδ

since from equation (5)

MOE Random = e

μ

 SGα

MOE Pa = MOE:Rδ

MOE Random

ln MOE Pa

MOE Random = δ ln MOE:R

δ =

 ln MOE Pa – ln MOE Random

ln MOE:R

Derivation of (MOE Pa)(1-d ) (MOE Pe)d

 = MOE

Random

Appendix D

from equation (11)

(C1)

MOE Pa = e

μ

 SGα MOE:Rδ

(D1)

From equation (5)

(C2) substitute

MOE Random = e

μ

 SGα

(D2)

(C3) MOE Pa = MOE Random (MOE:R)δ

(D3)

(C4)

since

(C5)

MOE Pa MOE:R = MOE Pe (D4)

MOE Pa

(MOE Pa)δ = MOE Random

(MOE Pe)δ

(MOE Pa)(1 – δ) (MOE Pe)

δ = MOE Random 

9

(D5)

(D6)
Appendix E

Examples of Adjustment of Test Data for

Specific Gravity (SG) and Flake Alignment

Given: a random board and an aligned board made from the same

flakes and having the following properties

MOE Random = 600,000 lb/in.2

SG Random = 0.719

MOE Pa = 1,000,000 lb/in.2

MOE Pe = 170,000 lb in.2

SG Aligned = 0.616

Problem: To adjust data to SG of 0.650 and alignment of

MOE:R = 5

Set μ = 13.73 (E1)

Then, from Equation (13)

(E2)

(E3)

(E4)

Note: if two random boards were available, μ and α could have

been calculated using equations (8) and (10).

2.0.2/86

10

Therefore, using equation (5)

MOE Random = e

μ

 SGα

(E5)

MOE RandomsG 0.616 = e

13.73 (0.616)1.29 = 491,385 lb/in.2

(E6)

and

MOE RandomSG 0.650 = e

13.73 (0.650)1.29 = 526,649 lb/in.2

(E7)

from equation (14)

(E8)

(E9)

Therefore, using equation (11)

MOE Pa = e

μ

 SGα MOE:Rδ

(E10)

@SG = 0.650 and MOE:R = 5

MOE Pa = e

13.73 (0.650)1.29 (5.0)0.4 = 1,002,557 lb/in.2

(E11)

and

MOE Pe = e

13.73 (0.650)1.29 (5.0)(0.4–1) = 200,511 lb/in.2

(E12)

U. S. GOVERNMENT PRINTING OFFICE:1986/641-044/20027