Nondestructive 

Assessment of 

Single-Span Timber 

Bridges Using a 

Vibration-Based Method

Xiping Wang

James P. Wacker

Angus M. Morison

John W. Forsman

John R. Erickson

Robert J. Ross

United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Paper

FPL–RP–627

In cooperation

with the

United States

Department of

Transportation

Federal

Highway

Administration
Abstract 

This paper describes an effort to develop a global dynamic 

testing technique for evaluating the overall stiffness of timber bridge superstructures. A forced vibration method was 

used to measure the natural frequency of single-span timber 

bridges in the laboratory and field. An analytical model 

based on simple beam theory was proposed to represent the 

relationship between the first bending mode frequency and 

bridge stiffness (characterized as EI product). The results 

indicated that the forced vibration method has potential for 

quickly assessing the stiffness of the timber bridge superstructure. However, improvements must be made in the 

measurement system to correctly identify the first bending 

mode frequency in bridges in the field. The beam theory 

model was found to fit the physics of the superstructure of 

single-span timber bridges and could be used to correlate 

first bending frequency to global stiffness if appropriate 

system parameters are identified. 

Keywords: timber bridges, NDE, vibration 

June 2005 

Wang, Xiping; Wacker, James P.; Morison, Angus M.; Forsman, John W.; 

Erickson, John R.; Ross, Robert J. 2005. Nondestructive assessment of 

single-span timber bridges using a vibration-based method. Res. Pap. 

FPL-RP-627. Madison, WI: U.S. Department of Agriculture, Forest Service, 

Forest Products Laboratory. 15 p. 

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Acknowledgments 

We express our sincere appreciation to Steve Popkowski and 

Ralph Miller of Ottawa National Forest and Jim Manderfield 

of Houghton County for providing the field bridges for this 

study. We would also like to thank Bob Vatalaro and Brian 

Brashaw at Natural Resources Research Institute of University of Minnesota Duluth for assistance with field testing 

work. 

This research project was partially funded through a cooperative research agreement between Michigan Technological 

University, the USDA Forest Service Forest Products 

Laboratory, and the Federal Highway Administration. 

Contents 

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

Analytical Model .................................................................. 2

Beam Theory..................................................................... 2

Plate Theory...................................................................... 3

Experimental Methods .......................................................... 4

Timber Bridge Structures.................................................. 4

Moisture Content Determination....................................... 4

Forced Vibration Testing .................................................. 4

Static Load Testing ........................................................... 4

Estimation of Bridge Weight ............................................ 5

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

Physical Characteristics of Timber Bridges..................... 5

Model Validation .............................................................. 6

Measured Natural Frequency............................................ 6

Measured Bridge Stiffness................................................ 7

Prediction of Bridge Stiffness........................................... 8

Conclusions........................................................................... 9

Future Research .................................................................... 9

Literature Cited ..................................................................... 9

Appendix A—Bridge Information Summaries ................... 10

Stony Creek Bridge......................................................... 10

 Dead Stream Bridge ........................................................ 11 

East Branch Ontonagon River Bridge............................. 12

Jumbo River Bridge ........................................................ 13

Beaver Creek Bridge....................................................... 14

Appendix B—Summary of Static Load Test Data.............. 15
Nondestructive Assessment of 

Single-Span Timber Bridges Using 

a Vibration-Based Method 

Xiping Wang, Research Associate 

University of Minnesota–Duluth and Forest Products Laboratory, Madison, Wisconsin

James P. Wacker, Research General Engineer 

Forest Products Laboratory, Madison, Wisconsin

Angus M. Morison, Former Graduate Student 

Department of Mechanical Engineering, Michigan Technological University, Houghton, Michigan

John W. Forsman, Assistant Research Scientist 

School of Forest Resources and Environmental Science, Michigan Technological University, Houghton, Michigan 

John R. Erickson, Director (retired) 

Robert J. Ross, Project Leader 

Forest Products Laboratory, Madison, Wisconsin 

Introduction 

The deterioration of wood structures often occurs internally 

before any external signs of damage appear. The loadbearing capacity of the affected member is greatly reduced 

before surface deterioration is visible. Determination of an 

appropriate load rating for an existing structure and rational 

decisions about rehabilitation, repair, or replacement can be 

made only after an accurate assessment of the existing condition. Knowledge of the condition of a structure can reduce 

repair and replacement costs by minimizing labor and 

materials and extending service life. 

In general, structural condition assessment requires the 

monitoring of some indicating parameters that are sensitive 

to the damage or deterioration mechanism in question. 

Current inspection methods for wood structures are limited 

to evaluating each structural member individually, which is a 

labor-intensive, time-consuming process. For field assessment of wood structures, a more efficient strategy would be 

to evaluate structural systems or subsystems in terms of their 

overall performance and serviceability. From this perspective, examining the dynamic response of a structural system 

might provide an alternative way to gain insight into the 

ongoing performance of the system. Deterioration caused by 

an organism or physical damage to the structure reduces the 

strength and stiffness of the materials and thus could affect 

the dynamic behavior of the system. For example, if a structural system or section of the system were found to respond 

to dynamic loads in a manner significantly different from 

that observed in previous inspections, this would warrant a 

more extensive inspection of the structure. 

Recent cooperative research efforts of the USDA Forest 

Service Forest Products Laboratory, Michigan Technological 

University, and University of Minnesota–Duluth have resulted in significant progress in developing global dynamic 

testing techniques for nondestructively evaluating the structural integrity of wood structure systems. In particular, a 

forced vibration response system was developed and used to 

assess the global stiffness of wood floor systems in buildings 

(Soltis and others 2002, Ross and others 2002, Wang and 

others 2005). In these studies, a series of laboratoryconstructed wood floor systems and several in-place wood 

floor structures were examined. An electric motor with an 

eccentric rotating mass was built and attached to the floor 

decking to excite the structure. The response of the floor to 

the forced vibration was measured at the bottom of the joists 

using a linear variable differential transducer (LVDT). 

The damped natural frequencies of the floor systems were 

identified by increasing motor speed until the first local 

maximum deflection response was observed. The period 

of vibration was then estimated from the cycles of this 

steady-state vibration. 

The forced vibration approach was investigated for two 

reasons. First, compared with other techniques, the simplicity of this technique requires less experimental skill to perform field vibration testing. This fits the need of field inspectors who usually have little advanced training in 

structural dynamic testing. Second, the cost of testing a 

structure using the forced vibration method is very low 

compared with the use of a modal testing method. 

Furthermore, because forced vibration is a pure time 

domain method, it eliminates the need for knowledge of 

modal analysis. 
2 

Results from previous experimental studies showed that 

vibration generated through a forcing function could enable 

a stronger response in wood floor systems and give consistent frequency measurement. A decrease in natural frequency seems proportionate to the amount of decay, as 

simulated by progressively cutting the ends of joists in laboratory floor settings (Soltis and others 2002). In addition, the 

analytical model derived from simple beam theory was 

found to fit the physics of floor structures and can be used 

to correlate the natural frequency (first bending mode) to 

EI product of the floor cross section (Wang and others 

2005). 

Cooperative research to date has provided a reasonable 

scientific base upon which to build an engineering application of vibration response as part of a wood structure inspection program. The purpose of this study is to extend global 

dynamic testing methods, specifically the forced vibration 

testing technique, to timber bridges in the field. It is to be 

used as a first pass method, identifying timber bridges that 

need more thorough inspection. To simplify the method as 

much as possible (in regard to field application), we focus on 

only the first bending mode of the bridge vibration. Specifically, we correlate the frequency of the first bending mode 

to the stiffness characteristics of single-span girder-type 

timber bridges. 

Analytical Model 

The fundamental natural frequency was chosen as the indicator of global structure stiffness. For the purpose of practical 

inspection, an analytic model is needed for relating the 

fundamental natural frequency to the global stiffness properties of a bridge. Continuous system theory was chosen as the 

means for developing an analytical model that is based on 

general physical properties of bridges, such as length, mass, 

and cross-sectional properties. Specifically, continuous beam 

and plate theories were investigated as bases for the model, 

providing two initial relationships between bending mode 

frequency and EI product. 

Beam Theory 

The superstructure of a single-span timber girder bridge is 

typically constructed of wood beams (stringers), cross 

bridging, deck boards, and a railing system (Fig.1). The 

beam theory model is based on the modal response of a 

single beam of rectangular cross section. To relate a singlespan wood girder bridge to the beam theory, we assume that 

the stiffness of the stringers predominates over that of the 

transverse deck sheathing and provides all the stiffness. 

The total mass of the deck and railing system is distributed 

into the assumed mass of the stringers. This assumption is 

made because the thickness of the decking boards is relatively small compared with the height of the stringers. 

In addition, the deck is not continuous and the deck boards 

are nailed perpendicular to the stringers, reducing the stiffness that would be provided in the case of simple bridge 

bending. The cross bridging also does not contribute to the 

bending stiffness of the bridge because it mainly provides 

lateral bracing to the beams. Under this assumption, a singlespan wood girder bridge behaves predominately like a beam 

with resisting moments in the vertical direction. 

Figure 1—Typical single-span girder timber bridge: 

(a) side view, (b) structure, (c) deck. 
3

The partial differential equation (PDE) governing vertical 

vibration for a simple flexure beam is 

0 4

4

2

2

= ∂

∂

⎟

⎠

⎞ ⎜

⎝

⎛ +

∂

∂

x

u

m

EI

t

u (1) 

where 

u is vertical displacement variable, 

t time, 

x variable along beam length, 

E modulus of elasticity, 

I area moment of inertia of bridge cross section, and 

m mass per unit length. 

The solution of this PDE is generally accomplished by 

means of the separation of variables and is largely dependent 

on boundary conditions at each end of the beam. Blevins 

(1993) showed that a general form for the natural frequency 

for any mode can be derived as 

1 2

2

2

2

⎟

⎠

⎞ ⎜

⎝

⎛ = m

EI

πL

λ

f i

i (2) 

where 

fi is natural frequency (mode i), 

λi a factor dependent on beam boundary conditions, 

L beam span, 

I moment of inertia, and 

EI stiffness of beam. 

Equation (2) ignores the contributions of rotary inertia and 

shear in the beam cross section. The more complicated PDE 

including these factors is 

0 4

4

2 2

4

2

2

4

4

=⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

⋅ ⋅

⋅ +⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂ ∂

∂ ⎟

⎠

⎞ ⎜

⎝

⎛

⋅ ⋅

⋅ − +

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

t

u

k A G

J m

x t

u

k A G

EI m J

t

u

m

x

u EI

 (3) 

where 

J is mass moment of inertia of bridge cross section, 

k shear coefficient, 

A cross-sectional area, and 

G modulus of rigidity. 

Close form solutions for this PDE are not generally available. In analyzing the relative importance of added factors to 

natural frequencies, Morison (2003) found that for typical 

timber bridge properties, the effect on frequency is only 

about 1%. 

Plate Theory 

Considering the geometric form of its cross section, a singlespan girder timber bridge may be modeled as a ribbed plate 

(Fig. 2). Here, the assumption is that the deck is continuously and perfectly attached to the stiffening ribs. The general expression for the natural frequencies of an orthotropic 

plate is (Blevins 1993) 

1 2

2 2

) 1 2 1 2 4 (

2

2γ 2 2

1 2

4

4

2

4

4

1

1 2

⎥

⎥

⎦

− ⎤

+

⎢

⎢

⎣

⎡

= + +

a b

J J H H k D

a b

H H D

b

G D

a

G D f x y xy

ij

π

 (4) 

where 

i and j are mode number indexes, 

γ is mass per unit area of deck, 

G1, G2, J1, J2, H1, and H2 are system parameters, 

D is a cross-sectional constant, 

a is bridge length, and 

b is bridge width. 

The system parameters in Equation (4) with a subscript of 1 

refer to the width dimension of the bridge (constrained 

edges), and a subscript of 2 relates to the length dimension 

(free edges). For a simple bending mode with elementary 

boundary conditions, many system parameters become zero, 

allowing for great simplification of Equation (4): 

1 2

1

4

r 4

1 2 2 ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ = L a

G EI fi

γ

π (5) 

where 

Ir is area moment of inertia of reduced cross-section, 

L length of bridge, and 

a1 stringer spacing. 

Figure 2—Ribbed plate cross-section. 
4 

In this model, the total mass of the system is distributed over 

the surface area of the bridge deck and the area moment of 

inertia is calculated for a subsection of the overall bridge 

cross-section, as illustrated in Figure 2. 

Experimental Methods 

Timber Bridge Structures 

Two experimental bridges were constructed in laboratory 

settings so that controlled experiments could be conducted. 

The first laboratory bridge (designated as Lab 1) was actually a field bridge (Onion Creek Bridge) that was removed 

from service and relocated to the laboratory. The bridge 

measured 9 ft (2.7 m) wide by 16 ft (4.9 m) long, and the 

superstructure consisted of six 6- by 12-in. by 16-ft. (15.2- 

by 30.5-cm by 4.9-m) stringers and 3- by 10-in. (7.6- by 

25.4-cm) deck boards and running planking. The second 

laboratory bridge (designated as Lab 2) measured 9 ft 

(2.7 m) wide by 21 ft (6.4 m) long. It was built with six 

6- by 12-in. (15.2- by 30.5-cm) Douglas-fir and eastern 

white pine timbers and 3- by 8-in. (7.6- by 20.3-cm) plank 

deck with known material properties. Both laboratory 

bridges were rested on 12- by 12-in. (30.5- by 30.5-cm) sill 

plates anchored to the floor with angle iron, which approximated a simply supported boundary condition. 

In addition to laboratory bridges, five field timber bridges 

currently in service on the Kenton Ranger District of the 

Ottawa National Forest in the Upper Peninsula of Michigan, 

all of similar design (timber stringer plus plank deck), were 

examined in October 2002 (Table 1). All these bridges were 

built in the early 1950s, and their initial designs were based 

upon American Association of State Highway Transportation Officials (AASHTO) standard truck loading. Bridge 

length measured on site ranged from 20 to 44 ft (6.1 to 

13.4 m) (out–out) and width from 15 to 16 ft (4.6 to 4.9 m) 

(out–out). The superstructure of each bridge span consisted 

of 10 creosote-treated sawn lumber stringers (6- by 12-in. 

(15.2- by 30.5-cm) to 6- by 16-in. (15.2- by 40.6-cm)), with 

3-in.- (7.6-cm-) thick transverse plank decks nailed perpendicular to the stringers. Running planks nailed in two strips 

parallel to the direction of the stringers served as a wearing 

surface for the single-lane bridges. Details for each field 

bridge are summarized in Appendix A. 

Moisture Content Determination 

At the time of bridge testing, wood moisture content of each 

bridge was measured with an electrical-resistance-type moisture meter and 3-in.- (76-mm-) long insulated probe pins in 

accordance with ASTM D 4444 (ASTM 2000). Moisture 

content data were collected at pin penetrations of 1, 2, and 

3 in. (25, 51, and 76 mm) from the underside (tension face) 

of three different timber beam girders for each bridge. All 

field data were corrected for temperature adjustments in 

accordance with Pfaff and Garrahan (1984). 

Forced Vibration Testing 

A forced vibration technique was used to identify the first 

bending mode frequency of the bridge structures. This 

method is a purely time domain method and was proposed 

because it eliminates the need for modal analysis. An electric 

motor with a rotating unbalanced wheel is used to excite the 

structure (Fig. 3), which creates a rotating force vector proportional to the square of the speed of the motor. Placing the 

motor at midspan ensured that the simple bending mode of 

structure vibration was excited. A single piezoelectric accelerometer (PCB U353 B51), also at midspan, was used to 

record the response in the time domain. To locate the first 

bending mode frequency, the motor speed was slowly increased from rest until the first local maximum response 

acceleration was located. The period of vibration was then 

estimated from 10 cycles of this steady-state motion. 

Static Load Testing 

Because the primary goal of this work was to relate the 

vibrational characteristics of the timber bridge structures to a 

measure of structural integrity, the bridges were also evaluated with the established method of static load–deflection 

field testing. Stiffness of bridge superstructure (EI product) 

could then be estimated from the field test results. 

Static load tests were conducted on each field bridge using a 

live load testing method. A test vehicle was placed on the 

bridge deck, and the resulting deflection was measured from 

calibrated rulers suspended from each timber girder along 

the midspan cross-section using an optical surveying level 

Table 1—Summary information for field timber bridgesa

Live load 

Bridge 

Bridge 

length,

out–out

(ft) 

Bridge 

width, 

out–out 

(ft) 

Simple 

spans 

(no.) 

Initial 

design 

Current 

posting 

Year 

built 

Stony Creek 20 15 1 HS20 15 ton 1954 

Dead Stream 44 15 2 HS20 20 ton 1954 

E.B.O. River 26 15 1 H15 None 1950 

Jumbo River 24 16 1 H15 None 1950 

Beaver Creek 43 15 2 H15 None 1954 

a

1 ft = 0.3048 m. bEast Branch Ontonagan River 
5

(Fig. 4). The test vehicle consisted of a fully loaded, tri-axle 

gravel truck with a gross vehicle weight of 47,740 lb 

(212.37 kN) (individual axle weights of 13,420, 17,160, and 

17,160 lb (59.70, 76.33, and 76.33 kN)). The rear 17,160-lb 

(76.33-kN) axles were spaced 4.4 ft (1.3 m) apart, and the 

13,420-lb (59.70-kN) axle was 13.4 ft (4.1 m) from the rear 

axles (Fig. 5). Deflection readings were recorded prior to 

testing (unloaded), after placement of the test truck for each 

load case (loaded), and at the conclusion of testing (unloaded). For each load test, the test vehicle straddled the 

bridge centerline; the bridge midspan bisected the rear dual 

truck axles. Figure 5 details the loading condition for the 

static test. Measurement precision was ±0.04 in. (±1.0 mm) 

with no vertical movements detected at the bridge supports. 

The static EI product of each bridge was estimated from 

load–deflection data on the basis of conventional 

beam theory. 

Estimation of Bridge Weight 

As known from the theoretical model shown in Equations (2) and (5), bridge weight is needed in predicting structural stiffness using the vibration response method. In this 

study, bridge weights were estimated based on actual dimensions along with an estimated unit weight for timber components. A conservative unit weight of 50 lb/ft3

 (801 kg/m3) is 

required for computing dead loads in the design of timber 

bridges according to AASHTO Standard Specifications for 

Highway Bridges. A less conservative unit weight of 

35 lb/ft3

 (561 kg/m3), which may more closely represent the 

actual density of creosote-treated Douglas-fir bridge components, was assumed in computing bridge weights for all five 

field bridges. Douglas-fir was most likely the wood species 

because visual evidence of incising typically associated 

with Douglas-fir (and other difficult-to-treat species) was 

observed at all field bridges. 

Results and Discussion 

Physical Characteristics 

of Timber Bridges 

The physical characteristics of the two laboratory bridges 

and five field timber bridges are summarized in Table 2. Of 

the five field bridges tested, two (Dead Stream and Beaver 

Creek) actually consisted of two spans. Because access to 

the bridge underside of these two bridges did not permit 

static load testing on both spans, we did field testing on only 

one span for each bridge. The tested bridge spans of these 

two bridges were treated as single-span bridges. The span 

length of tested bridges therefore ranged from 16 to 24.2 ft 

(4.9 to 7.4 m). 

Figure 3—Forced vibration testing of field bridges 

with a forcing function. 

Figure 4—Static load testing of field bridges with 

fully loaded gravel truck. 

Figure 5—Schematic of static loading pattern: 

(a) top view, (b) side view. 
6 

The moisture content of the field bridge timbers ranged from 

13% to 25%, but that of most timbers was below 20%. 

Average moisture content was below 20%, with the exception of the Jumbo River Bridge, where some beams had a 

moisture level of 23% to 25%. 

Model Validation 

To select the most appropriate model, a validation process 

was performed through testing a laboratory-constructed 

timber bridge. The experimental bridge setup designated as 

Lab 2 was selected because a known value of the boundary 

condition factor exists for this setup. This allowed 

Figure 6—Model predictions and measured data for 

laboratory bridge.

comparison of the model’s prediction with the actual measured performance of the structure. Figure 6 shows the results 

of the validation process. Each candidate model was used to 

predict the relationship between EI product and bending 

mode frequency for the bridge, using the known values of 

mass and the physical dimensions of the system. The data 

points represent the measured performance of the bridge. It 

is evident from this figure that beam theory is the obvious 

choice to model bridge behavior. 

The error between the beam theory predictions and measured 

frequencies was less than 3%. The huge overprediction of 

frequency with the plate model points to one of the largest 

discrepancies between the plate model and the physical 

structure. The deck is a segmented structure in reality, 

whereas the plate model assumes that the deck is continuous. 

Because the deck is segmented, it does not carry much of the 

bending moment and therefore contributes little to the bending stiffness of the bridge structure. The beam model, on the 

other hand, matches this assumption and best captures the 

physics of this type of timber bridge. Therefore, the simple 

beam model was proposed to represent the relationship 

between the first bending mode frequency and bridge 

stiffness. 

Measured Natural Frequency 

To verify the bending mode frequency measured by the 

forced vibration method, modal testing results of the bridges 

were obtained. Modal testing and analysis were conducted in 

a parallel study focused on investigating the use of impactgenerated frequency response functions (FRFs) for bridge 

Table 2—Physical characteristics and measured natural frequency of timber bridgesa

Measured bending 

mode frequency (Hz) 

Bridge 

Bridge 

spanb

(ft) 

Size of stringers 

(in.) 

Avg. moisture 

content of 

stringersc

(%) 

Forced 

vibration 

Modal 

analysis 

Laboratory bridges 

 Lab 1 (Onion Creek) 16 6 by 12 n/a 19.50 19.70 

 Lab 2 (Sands) 21 6 by 12 22 9.63 9.82 

Field bridges 

 Stony Creek 18.5 5.25 by 13 18 17.85 23.75 

 Dead Stream (1 span) 21.5 5.38 by 15.25 15.5 17.70 17.96 

 E.B.O. River 24.0 6 by 16 18.9 12.30 14.43 

 Jumbo River 24.2 5.88 by 15.75 21.8 14.15 13.37 

 Beaver Creek (1 span) 21.5 5.50 by 15.25 18.2 19.90 17.31 

a

1 ft = 0.3048 m, 1 in. = 2.54 cm. 

b

Only one span was tested for Dead Stream Bridge and Beaver Creek Bridge. 

 The tested span length of these two bridges was half the total span length. c

Average of moisture content readings collected at pin penetration 3-in. (76-mm) 

 deep from underside (tension face) of three timber beam girders. 
7

evaluation (Morison 2003, Morison and others 2002). The 

results of both forced vibration frequency and first bending 

mode frequency identified by modal analysis technique are 

shown in Table 2. 

Figure 7 compares measured natural frequency from forced 

vibration testing and bending mode frequency from standard 

modal testing. It shows that under laboratory conditions, 

frequencies measured using the forced vibration method 

matched quite closely (less than 2% difference) the bending 

mode frequencies found from modal analysis techniques. In 

the field, however, much more disparity was noticed. The 

difference in frequencies for the Jumbo River and Dead 

Stream bridges was 5% or less. These results correspond to 

the bridges for which the bending mode was the lowest 

mode, and this mode was clearly separated from other modes 

of the structure (Morison 2003). Error for the Stony Creek 

 

Figure 7—Comparison of measured natural frequency 

from forced vibration method and first bending mode 

frequency from standard modal testing. 

and E.B.O. River bridges was larger because the forced 

vibration results corresponded to a mode other than the 

bending mode. The modal analysis indicated that the 3rd 

mode frequency for the Stony Creek bridge and the 2nd 

mode frequency for the E.B.O. River bridge were actually 

the true bending mode. It seems that the closely spaced 

modes of the bridges made finding the peak amplitude of the 

bending mode difficult. This could have been caused by the 

lack of uniformity for the boundary condition and material 

properties across the width of the bridges and the much 

higher modal density of these bridges compared with that of 

the other bridges tested. 

Measured Bridge Stiffness 

Table 3 shows static live load deflections from field load 

tests. No permanent deflection was noted on the stringers 

monitored, and no support movements were detected during 

truck loading tests. Because of the nature of the bridge structure and live truck loading, the deflection of stringers was 

uneven across the width of each bridge. Maximum deflection typically occurred in the center stringers, ranging from 

0.386 in. (9.8 mm) for Beaver Creek Bridge to 0.551 in. 

(14.0 mm) for E.B.O. River Bridge. The average live load 

deflection ranged from 0.272 in. (6.9 mm) for Stony Creek 

Bridge to 0.413 in. (10.5 mm) for E.B.O. River Bridge. The 

load–deflection curves of the filed bridges are shown in 

Appendix B. 

To correlate the first bending mode frequency to load testing 

results, the following assumptions were made to compute the 

EI product of the bridges: 

1.The superstructure of timber bridges is similar to a beamlike structure with symmetrically placed loads. 

2.The bridges are close to being simply supported (for the 

purpose of static deflection analysis only). 

3.The average deflection of each bridge is equivalent to the 

value that characterizes the deflections of all stringers if 

the load is applied evenly across the width of the bridge. 

Given these assumptions, the static beam deflection theory 

provides the following relationship for a beam-like structure: 

(3 4 ) 24δ

2 2 L a

Pa EI = − (6)

where 

P is static load of individual axle, 

δ average midspan deflection, 

L span length of bridge, and 

a distance from bridge support to nearest loading point. 

The calculated EI products of the field bridges are shown in 

Table 3 and are hereafter referred to as the measured EI 

Table 3—Summary of static load testing deflections 

and measured bridge stiffnessa

Bridge 

Maximum 

live load 

deflection 

(in.) 

Average 

live load 

deflection 

(in.) 

Measured 

bridge stiffness

(EI product) 

(×106

 lb-in2) 

Stony Creek 0.413 0.272 26,268 

Dead Stream 0.394 0.280 34,801 

E.B.O. River 0.551 0.413 42,369 

Jumbo River 0.433 0.346 38,708 

Beaver Creek 0.386 0.276 32,665 

a

1 in. = 2.54 cm, 1 lb-in2 = 2.87 × 103 N-m2. 
8 

because they were derived from measured static data. 

Based on static load testing results, the structural stiffness 

(EI product) of the field bridges ranged from 

26,268 ×106

 lb-in2 (75.39 ×106 N-m2) (Stony Creek Bridge) 

to 42,368 ×106

 lb-in2 (121.60 ×106 N-m2) (E.B.O. 

River Bridge). 

Prediction of Bridge Stiffness 

Equation (2) is a general form for the natural frequency of 

any mode for a simply supported beam. If vibration is restricted to the first mode, Equation (2) can be rearranged to 

obtain an expression for stiffness (EI) as 

2 3

1

1 f WL

Kg

EI ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = (7) 

where 

f1 is fundamental natural frequency (first bending mode), 

K system parameter dependent on beam boundary conditions (pin–pin support, K = 2.46; fix–fix support, K= 

12.65), 

W weight of beam (uniformly distributed), and 

g acceleration due to gravity. 

Figure 8 shows the theoretical prediction for two extreme 

supporting conditions (free–free and fixed–fixed) and experimental data obtained from field bridges. Here, EI/WL3

 is 

treated as the independent variable and natural frequency as 

the dependent variable. The natural frequency is predicted 

over a range of EI/WL3

 assuming both simply supported and 

fixed boundary conditions. The measured data are then 

superimposed on the same set of axes. It is noted that measured results lie between simple support and rigidly fixed 

boundary conditions, with a bias toward the simply supported prediction. To characterize the boundary condition of 

each bridge, a system parameter K was determined based on 

experimental data of the field bridges. The average system 

parameter that best described all field bridges tested was 

found to be 4.20, with a standard deviation of 0.690. 

With the newly developed system parameter K, the model in 

Equation (7) could be used to predict the EI product of 

bridges using measured natural frequency. Figure 9 compares the predicted EI product from the forced vibration 

method and measured EI product from static load testing. 

Although the EI predictions for the Dead Stream and Jumbo 

River Bridges were quite close to measured EI (less than 7% 

difference), the overall performance of the prediction model 

suffers from a significant error. It appears that prediction of 

bridge stiffness has a significant variation, from 2% minimum to 37% maximum difference (in absolute value). 

Figure 8—Theoretical predictions and experimental data. 

Figure 9—Comparison of predicted El and measured El

of field brides. 

Several factors contributed to this prediction error. First, one 

source is obviously the forced vibration method itself. As 

indicated in the previous discussion, the estimates of first 

bending mode frequency from forced vibration testing 

contained significant errors in some cases. The error was a 

direct result of the bending mode not being the lowest in 

natural frequency, so that other modes (typically torsion) 

were misidentified as the bending mode. In the case where 

first bending mode frequency was properly identified (such 

as for the Dead Stream and Jumbo River Bridges), the predicted EI showed much less difference from measured EI. 

The second error source in EI prediction is most likely the 

inaccurate estimate of bridge weight. Bridge weight information is essential in calculating EI product based on beam 

theory model. In this study, bridge weights were estimated 

based on actual dimensions along with an estimated unit 

weight for timber components. The true wood density of 

each bridge might be significantly different from the 
9

assumed unit weight. Other factors could also affect bridge 

weight, which make estimation difficult (such as species and 

moisture difference, wood deterioration, dirt or debris collected on the deck). 

Third, in spite of structural similarities, the boundary condition of each field bridge is unique due to the construction 

variability, load history, and road and soil conditions. The 

overall system parameter K used for EI prediction here is the 

average value of the system parameter Ki of each bridge, 

which describes the entire population. The small sample size 

(five field bridges) is therefore a contributing factor. If more 

field bridges had been available, a more representative average could have been obtained. 

Conclusions 

A forced vibration method was used to measure the natural 

frequency of single-span timber bridges in the laboratory and 

the field. An analytical model based on beam theory was 

proposed to represent the relationship between the first 

bending mode frequency and bridge stiffness characterized 

as EI product. From the results of this study, we conclude 

the following: 

• The forced vibration method has the potential to be used in 

the field to quickly assess timber bridge superstructure 

stiffness. However, improvements need to be made in testing procedure and measurement system to correctly identify the first bending mode frequency as a forcing function 

is applied. 

• Timber bridge weight is essential for predicting bridge 

stiffness based on the beam theory model. Weight estimation based on wood volume and estimated unit weight for 

timber components seems inadequate to obtain reliable 

results. 

• The analytical model generated from simple beam theory 

fits the physics of single-span girder bridges better than 

the model from pate theory, but more representative system parameters need to be developed to better correlate 

measured bending mode frequency to EI product. 

Future Research 

The experimental data collected from this study are still 

limited given the structural complexity of timber bridges in 

the real world. More analytical and experimental work is 

needed to fully understand the physics and structural conditions in terms of vibration response and load capacity. A 

new joint timber bridge research project is now underway at 

the University of Minnesota–Duluth and the USDA Forest 

Service, Forest Products Laboratory to further investigate 

key issues in vibration modes and boundary conditions and 

to refine field testing techniques and instrumentation systems. More field timber bridges in northern Minnesota with 

various structural conditions will be tested with improved 

forced vibration response methods. To eliminate or reduce 

error in estimating the bending mode frequency, two accelerometers will be placed on opposite sides of the bridge 

superstructure. Bending mode frequency will be determined 

by examining both maximum acceleration and phase information from two simultaneous vibration response signals. 

Field vibration measurements will also be coupled by condition evaluation using traditional inspection techniques and 

standard live load tests using a loaded truck. To improve the 

reliability of vibration response methods, more comprehensive mathematical models will be developed to quantify the 

sensitivity of bridge response to various environmental, 

experimental, and architectural factors. 

Literature Cited 

ASTM. 2000. Use and calibration of hand-held moisture 

meters. ASTM D 4444–92. Philadelphia, PA: American 

Society for Testing and Materials. 

Blevins, R.D. 1993. Formulas for natural frequency and 

mode shape. Malabar, Florida: Krieger Publishing Company. 

Morison, A.M. 2003. Nondestructive evaluation of short 

span timber bridges with impact generated FRFs. Houghton, 

MI: Michigan Technological University. M.S. Thesis. 

Morison, A.M.; Van Karsen, C.D.; Evensen, H.A. [and 

others]. 2002. Timber bridge evaluation: A global nondestructive approach using impact generated FRFs. In: Proceedings, 20th International Modal Analysis Conference, 

1567–1573. 

Pfaff, P.; Garrahan, P. 1984. Moisture content correction 

tables for the resistance-type moisture meter. Eastern species. SP511E. Ottawa, Ontario, Canada: FORINTEK 

Corporation. 37 p. 

Ross, R.J.; Wang, X.; Hunt, M.O.; Soltis, L.A. 2002. 

Transverse vibration technique to identify deteriorated wood 

floor systems. Experimental Techniques. 26(4): 28–30. 

Soltis, L.A.; Wang, X.; Ross, R.J.; Hunt, H.O. 2002. 

Vibration testing of timber floor systems. Forest Products 

Journal. 52(10): 75–81. 

Wang, X.; Ross, R.J.; Hunt, M.O.; Erickson, J.R. 2005. 

Low frequency vibration approach for assessing performance of wood floor systems. Wood and Fiber 

Science. 37(3). 
10 

Appendix A—Bridge Information Summaries 

Stony Creek Bridge 

Location 

 State: Michigan 

 County: Houghton County 

 Highway: Forest Road 3660 or County Route D162 

 Waterway crossing: Stony Creek 

 Section, Township, Range: S33, T47N, R36W 

Owner: Houghton County 

Year built: 1954 

Design configuration: Simply supported 

 Length: 20 ft 

 Width: 15 ft 

 No. spans: 1 

 No. traffic lanes: 1 

 Loading: AASHTO HS20–44 

Superstructure type: Sawn lumber stringer with transverse plank deck 

 No. stringers: 10 

 Stringer size: 5.25 by 13 in. 

 Spacing (c–c): 18 in. 

 Decking size: 3 by 8 in. 

 Species: Douglas-fir 

 Preservative: Creosote 

 Skew: 0°

Substructure type: Timber pile with timber cap 

Wearing surface: Timber running planks 

Recent condition rating: 2001 inspection rating of 7 for all components
11

Dead Stream Bridge 

Location 

 State: Michigan 

 County: Houghton 

 Highway: Forest Road 2400 or County Route D163C 

 Waterway crossing: Dead Stream 

 Section, Township, Range: S33, T47N, R35W 

Owner: Houghton County 

Year built: 1954 

Design configuration: Simple span 

 Length: 44 ft 

 Width: 15 ft 

 No. spans: 2 

 No. traffic lanes: 1 

 Loading: AASHTO HS20–44 

Superstructure type: Sawn lumber stringer with transverse plank deck 

 No. stringers: 10 

 Stringer size: 15.25 by 5.38 in. 

 Spacing (c–c): 18 in. 

 Decking size: 3 by 8 in. 

 Species: Douglas-fir 

 Preservative: Creosote 

 Skew: 0°

Substructure type: Timber pile with timber cap 

Wearing surface: Timber running planks 

Recent condition rating: 2001 inspection rating of 7 for all components 
12 

East Branch Ontonagon River Bridge 

Location 

 State: Michigan 

 County: Houghton 

 Highway: Forest Road 3500 

 Waterway crossing: East Branch Ontonagon River 

 Section, Township, Range: S26, T47N, R36W 

Owner: Ottawa National Forest 

Year built: 1950 

Design configuration: Simple span 

 Length: 26 ft 

 Width: 15 ft 

 No. spans: 1 

 No. traffic lanes: 1 

 Loading: AASHTO H15–44 

Superstructure type: Sawn lumber stringer with transverse plank deck 

 No. stringers: 10 

 Stringer size: 6 by 16 in. 

 Spacing (c–c): 18 in. 

 Decking size: 3 by 8 in. 

 Species: Douglas-fir 

 Preservative: Creosote 

 Skew: 0°

Substructure type: Timber pile with timber cap 

Wearing surface: Timber running planks (size) 

Recent condition rating: 2002 ratings—deck (5), superstructure (6), substructure (5) 
13

Jumbo River Bridge 

Location 

 State: Michigan 

 County: Iron County 

 Highway: Forest Road 3610 

 Waterway crossing: East Branch Jumbo River 

 Section, Township, Range: S10, T46N, R37W 

Owner: Ottawa National Forest 

Year built: 1950 

Design configuration: Simple span 

 Length: 24 ft 

 Width: 16 ft 

 No. spans: 1 

 No. traffic lanes: 1 

 Loading: AASHTO H15–44 

Superstructure type: Sawn lumber stringer with transverse plank deck 

 No. stringers: 10 

 Stringer size: 5.88 by 15.75 in. 

 Spacing (c–c): 18 in. 

 Decking size: 4 by 6 in. 

 Species: Douglas-fir 

 Preservative: creosote 

 Skew: 0°

Substructure type: Timber pile with timber cap 

Wearing surface: Timber running planks 

Recent condition rating: 2002 ratings—deck (7), superstructure (6), substructure (6) 
14 

Beaver Creek Bridge 

Location 

 State: Michigan 

 County: Houghton 

 Highway: Forest Road 1100 

 Waterway crossing: Beaver Creek 

 Section, Township, Range: S29, T48N, R37W 

Owner: Ottawa National Forest 

Year built: 1954 

Design configuration: Simple span 

 Length: 43 ft 

 Width: 15 ft 

 No. spans: 2 

 No. traffic lanes: 1 

 Loading: AASHTO H15–44 

Superstructure type: Sawn lumber stringer with transverse plank deck 

 No. stringers: 10 

 Stringer size: 5.50 by 15.25 in. 

 Spacing (c–c): 18 in. 

 Decking size: 3 by 8 in. 

 Species: Douglas-fir 

 Preservative: Creosote 

 Skew: 0°

Substructure type: Timber pile with timber cap 

Wearing surface: Timber running planks 

Recent condition rating: 2000 ratings—deck (6), superstructure (6), substructure (5) 
15

Appendix B—Summary of Static Load Test Data 

-25 

-20

-15 

-10 

-5

0 

0 1 2 3 4 5 6 7 8 9 10 11

Beam no.

Deflection (mm) 

Stony Creek Bridge 

Dead Stream Bridge -25 

-20 

-15 

-10 

-5 

0

0 1 2 3 4 5 6 7 8 9 10 11

Beam no.

Deflection (mm)

-25 

-20

-15 

-10 

-5 

0

0 1 2 3 4 5 6 7 8 9 10 11

Beam no.

Deflection (mm)

Jumbo River Bridge 

-25 

-20 

-15 

-10 

-5 

0

0 1 2 3 4 5 6 7 8 9 10 11

Beam no.

Deflection (mm) 

Beaver Creek Bridge

E. B. O. River Bridge -25 

-20 

-15 

-10 

-5 

0

0 1 2 3 4 5 6 7 8 9 10 11

Beam no.

Deflection (mm)