ORIGINAL ARTICLE

J Wood Sci (2006) © The Japan Wood Research Society 2006DOI 10.1007/s10086-005-0754-3

Y. Kubojima (*) · H. Kato · M. TonosakiForestry and Forest Products Research Institute, P.O. Box 16,Tsukuba Norin Kenkyu Danchi-nai, Tsukuba 305-8687, JapanTel. +81-298-73-3211; Fax +81-298-74-3720e-mail: kubojima@ffpri.affrc.go.jp

H. OhsakiHokkaido Forest Products Research Institute, Asahikawa 071-0198,Japan

Yoshitaka Kubojima · Hisashi Ohsaki · Hideo KatoMario Tonosaki

Fixed–fixed flexural vibration testing method of beams for timber guardrails

Received: April 25, 2005 / Accepted: July 28, 2005

Abstract Resonance frequencies of beams with varioustypes of end supports were examined for flexural vibration.Rectangular beams with dimensions of 300 (L) ¥ 25 (R) ¥ 5or 10mm (T) were used as the test specimens. Variouscompressing stresses were applied to the parts around bothends of a test beam and flexural vibration tests were con-ducted. The measured resonance frequency started to in-crease from the resonance frequency of a beam with simplysupported ends and was stable around the resonance fre-quency of a beam with fixed ends as the compressing stressincreased. The stable resonance frequency was lower thanthe theoretical value because perfect fixation of a beam to apost was difficult. From these results, the temporal changein resonance frequency itself, rather than the stable reso-nance frequency, is effective to examine whether a beamhas enough strength as a guardrail.

Key words Boundary conditions · Fixed ends · Flexuralvibration test · Resonance frequency · Timber guardrail

Introduction

In 1998, timber guardrails were approved for use in nationaland prefectural roads if they could pass a crash test withcars and trucks. Various timber guardrails have been subse-quently developed and more than 50km of timber guard-rails are in use today.1,2

A method to estimate deterioration of wood used forguardrails, however, has not been established. Conse-quently, it is difficult to decide when the guardrails shouldbe replaced.

For this purpose, obtaining the changes in strength overtime of the wood for the timber guardrail is important.Although the strength can be measured accurately by re-moving the wood from a guardrail, this method is difficult toadopt because it takes much time. Hence, a testing methodby which the appropriate strength can be measured in thelumber fixed to a post of a guardrail is needed. Naturally,the test should be conducted without rupturing thespecimen.

Young’s modulus is one of the properties that is relatedto the strength and can be obtained without damaging thespecimen. One method for measuring Young’s modulus is aflexural vibration test.

The flexural vibration test has been widely used becauseof its simplicity. There have been several studies on itstesting method: for example, the supporting position of aspecimen, the distance between a specimen and a head of avibration generator,3 and gripping of a cantilever-typespecimen have been examined. However, these studies donot assume measurements in actual end conditions of abeam used as a guardrail but rather for typical end condi-tions: hinged, clamped, sliding, and free.

A beam for a timber guardrail is fixed to a post by a boltthrough a bracket (Fig. 1). Therefore, the end conditions ofthe beams used as timber guardrails are thought to be be-tween those of simply supported ends and fixed ones. Inactual structures, however, the resonance frequency ofbeams with simply supported ends is increased by 20%when taking the restriction by joints and the effect of floorjoists into account.4–6

On the other hand, the resonance frequency of a beamwith fixed ends is about twice as large as that of a beam withsimply supported ends, according to the Euler-Bernoullielementary theory on bending. Hence, end conditions of abeam must be understood to accurately measure theYoung’s modulus of wood lumber in actual structures. Weinvestigated matters that require attention in conductingthe fixed–fixed flexural vibration test.

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Theory

An elastically supported beam

According to the Euler-Bernoulli theory, the differentialequation for the bending of a beam is

EI

yx

Ay

t= + =∂

∂∂∂

4

4

2

2 0r , (1)

where E, I, y, x, r, A, and t are Young’s modulus, momentof inertia of a cross section, lateral deflection, distancealong the beam, density, cross-sectional area, and time,respectively.

Solving Eq. 1 gives us

y C x C x C x C x= + + +1 2 3 4cos sin cosh sinh ,l l l ln n n n (2)

where C1–C4 are constants, and

l

rn

n4

22

= ( )pf A

EI, (3)

where fn and n are resonance frequency and resonancemode number, respectively. Then, resonance frequency is

f

ml

EIAn

n=2

22p r, (4)

where mn = lnl and l is length.The first resonance mode is discussed in this study. For

symmetric modes, C2 = C4 = 0.When the ends of a beam are restricted by springs for

vibration in the y-direction (k1 and k3 in Fig. 2) and rotation(k2 and k4 in Fig. 2), the boundary conditions are

xl

EIy

xk y EI

yx

kyx

xl

EIy

xk y EI

yx

kyx

= - = - =

= = = -

Ï

ÌÔÔ

ÓÔÔ

2

2

3

3 1

2

2 2

3

3 3

2

2 4

: ,

: ,

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

(5)7,8

Substituting Eq. 2 into Eq. 5 under conditions of k1 = k3

and k2 = k4 gives

EIl

kl

EIl

kl

EIl

kl

EIl

kl

C

C

l l l l l l

l l l l l l

nn n

nn n

nn n

nn n

31

31

2 2

1

3

2 2 2 2

2 2 2 2

0

0

sin cos sinh cosh

cos sin cosh sinh.

- -

- - +

È

Î

ÍÍÍÍ

˘

˚

˙˙˙˙

È

ÎÍ

˘

˚˙ =

È

ÎÍ

˘

˚˙

(6)

When the solution of Eq. 6 is not C1 = C3 = 0, the deter-minant of the matrix is 0. Then, the following equation isobtained.

EIl

k EI kl

EIl

k EI kl

l l l l

l l l l

nn

nn

nn

nn

31 2

31 2

2 2

2 20

tan tanh

tanh tan .

-ÊËÁ

ˆ¯̃

+ÊËÁ

ˆ¯̃

+ -ÊËÁ

ˆ¯̃

+ÊËÁ

ˆ¯̃

=(7)

With both end parts of a specimen being supported byposts in our experiment, k1 is assumed to be •. Conse-quently, Eq. 7 is

tan tanh

m m EIk l

mn nn2 2

2

2

+ = - . (8)

If k2 = 0, Eq. 8 becomes

tan tanh .

m mn n

2 2+ = • (9)

Hence, m1 is p. This is the value for a simply supportedbeam.

If k2 = •, Eq. 8 becomes

tan tanh .

m mn n

2 20+ = (10)

Fig. 1. Timber guardrail

Fig. 2. A test beam supported elastically

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Hence, m1 is 4.730. This is the value for a beam with fixedends.

The m1 of Eq. 8 is plotted to k2 as shown in Fig. 3.

Timoshenko theory of bending

Flexural vibration is influenced by shear and rotary inertia.Timoshenko added these terms to the Euler-Bernoulli el-ementary theory of bending and developed the followingdifferential equation of bending:9

EI

yx

Ay

tI

sEG

yx t

s IG

yt

∂∂

∂∂

∂∂ ∂

∂∂

4

4

2

2

4

2 2

2 4

41 0+ - +ÊËÁ

ˆ¯̃

+ =r r r, (11)

where G is shear modulus and s = 1.18 is the shear deflectioncoefficient.10

When Eq. 11 is solved under the fixed–fixed condition,the resonance frequency fn can be written as follows:

f

pl

EIAn

n=2

22p r. (12)

The value of pn in Eq. 11 is obtained by the followingtranscendental equations:

tanh

tan

pB p A p

pB p A p

B p A p

B p A p

nt2

n t n2

nt2

n t n

t n4

t n2

t n t n

21

21

1

1

4

4 2

2

2 4 2

+ -

+ += -

+ +

+ -(13)

for symmetric modes, where

A

IAl

sEGt = +Ê

ËÁˆ¯̃2

12

, (14)

and

B

IAl

sEGt = - +Ê

ËÁˆ¯̃2

12 . (15)

Experimental

Specimens

To investigate wood species with various density, kiri (pau-lownia, Paulownia tomentosa Steud.), sugi (Japanese cedar,Cryptomeria japonica D. Don), akaezomatsu (Sakhalinspruce, Picea glehnii Mast.), shioji (ash, Fraxinus spae-thiana Lingelsh.), and shirakashi (white oak, Quercusmyrsinaefolia Blume) were used as specimens. The dimen-sions of each specimen were 300mm (longitudinal, L) inlength, 25mm (radial, R) in width, and 5 or 10mm (tangen-tial, T) in thickness.

The specimens were conditioned at 20°C and 65% rela-tive humidity for several months. The tests were conductedunder the same conditions.

Vibration test

To obtain the resonance frequency of the first mode bybending, flexural vibration tests were conducted by the fol-lowing procedure. An apparatus (End condition controllerKS-200 Takachiho Seiki) shown in Fig. 4 was used to pro-vide various end conditions. The regions of 25 (L) ¥ 25mm(R) from both ends were supported by the posts of theapparatus, the cross section of which was 25 ¥ 25mm. Byscrewing a bolt attached to a load cell, the test beam wascompressed. The compressing load was measured by theload cell and recorded by a data logger. The vibration wasexcited in the direction of the thickness at the center part bya hammer. Motion of the beam was detected by a micro-phone in the center part. The signal was processed througha fast Fourier transform (FFT) digital signal analyzer toyield high-resolution resonance frequencies.

A free–free flexural vibration test was also undertaken tomeasure the Young’s and shear moduli. A test beam was

Fig. 3. Solutions of Eq. 8 for various k2. Parameters used: Young’smodulus 12GPa, width 25mm, thickness 5mm, length 300 mm

Fig. 4. The vibration test under various end conditions

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suspended by two threads at the nodal positions of the free–free vibration corresponding to its resonance mode. Thevibration was excited and recorded by the above-describedmethod. The Young’s and shear moduli were calculatedwith the following regression method, which is called the“TGH method” in this article.

Goens11 approximated the solution of the Timoshenkodifferential equation (Eq. 11) by a Taylor series. Hearmon12

regarded the approximated equation of the solution as alinear equation. Using the Young’s moduli based on theEuler-Bernoulli elementary theory of bending of severalresonance modes, the regression is made and the Young’sand shear moduli can be obtained from the slope and inter-cept.12 By the TGH method, Young’s modulus (ETGH) andshear modulus can be calculated at the same time using onlythe flexural vibration test.

Substituting ETGH and mn = 4.730 (the frequency equa-tion of a beam with free ends is the same as Eq. 10) into Eq.4, the resonance frequency without the shear and rotaryinertia effects (f0) was obtained. On the other hand, theresonance frequency taking the shear and rotary inertiaeffects into account (fS) was calculated by solving Eq. 13with “Mathematica Ver. 3.0” software (Wolfram Research)using ETGH and the shear modulus, and then substitutingETGH and the obtained pn into Eq. 12.

Results and discussion

For all the wood species used for the experiments, the mea-sured resonance frequency increased rapidly early in theincreasing compression load process and approached thevalue of the fixed ends (Fig. 5). This trend was similar to them1–k2 relationship shown in Fig. 3. These results show thatthe measured resonance frequency, and consequently thespecific Young’s modulus, ran the risk of being smaller thanthe true value if the wood was not fixed completely to thepost of a guardrail. It is difficult to place and remove a loadcell or a torque meter on site. Therefore, the stable reso-

nance frequency should be recorded by increasing the loadfor fixing it.

However, it should be noted that the measured reso-nance frequency that was stable at larger compression stresswas smaller than f0: for example, the resonance frequency atthe maximum compression stress (fM) was about 95%(thickness: 5mm) and 85% (thickness: 10mm) of f0 (Table1). These discrepancies cannot be ignored, especially in thecase of thickness = 10mm: the specific Young’s modulus is0.852 = 0.72 of the true value.

The resonance frequency of the part of a beam withoutcompression stress is decreased by the shear and rotaryinertia effects in flexural vibrations when a beam is thickand resonance mode number is large.9 Thus, such effectswere examined. The fact that fS decreased by only 0.3%–1%and less than fM (Table 1) means that the shear and rotaryinertia effects in the part without the compression stresswere not serious.

Fig. 5. Changes in resonancefrequency with compression for5-mm-thick spruce (left) and10-mm-thick spruce (right)

Table 1. Ratio of measured resonance frequency to estimatedfrequency

Wood type fM/f0 fS/f0

Thickness: 5 mmPaulownia 0.957 0.997Cedar 0.959 0.998Spruce 0.952 0.997Ash 0.940 0.996Oak 0.941 0.997Thickness: 10 mmPaulownia 0.856 0.990Cedar 0.885 0.992Spruce 0.841 0.987Ash 0.803 0.984Oak 0.867 0.991

fM, The measured resonance frequency at the maximum compressionstress; fS, the resonance frequency taking the effect of shear deforma-tion and rotary inertia into account estimated by substituting ETGH andpn into Eq. 12; f0, the resonance frequency without the effect of sheardeformation and rotary inertia estimated by substituting ETGH and mn =4.73 into Eq. 4

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Next, the shear deformation and rotation at the com-pressed part was examined to investigate the poor fixation.Although the part of a specimen around a post apparentlydid not vibrate in the thickness direction because of com-pression, it would be difficult to restrict the shear deforma-tion and rotation perfectly: dividing a test specimen intomany elements like a finite element method (FEM) simula-tion, the shear deformation and rotation of elementsaround x = 0 in Fig. 6 will not be restricted sufficiently. Thelength–thickness plane of such an element will be shear-deformed and rotated easily during vibration in the thick-ness direction. In this case, the slope around the compressedpart is not 0 and k2 will not be •.

According to the theory described above, under the con-

dition of k1 = • and k2 π •,

m mn n

2 20+ πtanh in Eq. 10.

Hence, m1 is smaller than the value for a beam with fixedends (Fig. 3), thus, the resonance frequency decreases.

Assuming that the size of the element is the same, thereare more elements in the thickness direction in a thickspecimen than in a thin one. Therefore, the resonance fre-quency decreased more in the case of thickness = 10mmthan 5mm. When k2 π •, mn is also influenced by E, I, andl from Eq. 8 (see Fig. 7).

Although it is difficult to obtain the resonance frequencyaccurately from the fixed–fixed flexural vibration test men-tioned above, this testing method is nevertheless effectiveto maintain and control qualities of timber guardrails. Thistesting method is useful for the 100% inspection of woodenbeams for timber guardrails and the results dominate me-chanical properties.

Fig. 6. Deformation of an element at compressed parts

Fig. 7. Method for solving Eq. 8

Members of metal guardrails are manufactured in facto-ries so that their qualities meet the requirements of Ministryof Land, Infrastructure, and Transport of the Governmentof Japan, which means that all members are tested. Simi-larly, the 100% inspection will have to be conducted in thecase of timber guardrails. Testing all wooden beams directlyafter constructing timber guardrails enables the initial prop-erties and location of each beam to be obtained simulta-neously. Consequently, periodic inspections of timberguardrails will be simplified. For example, inspections canbe concentrated on beams with lower initial resonancefrequency.

A visual inspection is also important and can be per-formed simply, but the vibration test has particularadvantages over it. The resonance frequency reflects themechanical properties of the inner layers, unlike the thevisual inspection which can only provide information onthe surface. Results of the visual inspection may also bedifferent from worker to worker. Even if a wooden beamlooks deteriorated, it may not be necessary to replace abeam if the mechanical qualities fulfill the requirements.

Conclusions

The resonance frequency in timber guardrail beams be-tween the simply supported condition and fixed conditionwas obtained from flexural vibration tests. The results wereas follows:

1. The measured resonance frequency started to increasefrom the resonance frequency of a beam with simplysupported ends and approached that of fixed ends as the

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compressing stress increased. This trend was similar tothe m1–k2 relationship in Eq. 8.

2. The stable resonance frequency was lower than the theo-retical value because perfect fixation of a beam to a postwas difficult: it would be difficult to perfectly restrict theshear deformation and rotation of the part of a specimenaround a post.

3. Temporal change in the stable resonance frequency willbe effective to estimate the deterioration of woodenbeams used for timber guardrails.

Acknowledgments This work was financially supported by the “Studyon durability design and maintenance of timber guardrails and timbersound insulation walls” of the “Research Project for Utilizing Ad-vanced Technologies in Agriculture, Forestry, and Fisheries.” Wethank Dr. Hiroshi Yoshihara of Shimane University and Ms. MasamiYaguchi for their assistance.

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