Use of vibration techniques to determine the rotational
stiffness of timber joints
Paul Crovella 1, Dr. George Kyanka 2
Abstract A method for determining the rotational stiffness of timber joints using the natural frequency
of vibration of the beam is presented. Reviewed research shows that joint stiffness is
important for frame behavior when the relative value of joint stiffness exceeds the beam
flexural stiffness (α >1). Lab results show that vibration could predict joint stiffness of both
hardwood (within 17%) and softwood (within 16%) beams when restrained in a metal
fixture (α>1). A beam tested in a metal fixture with sequentially weakened joints shows
that the stiffness decreased with a matching decrease in frequency. Two softwood frames
with all timber joinery, one with tight and one with loose joints, were tested. Results show
that tight wood joinery can produce values of α>1. The stiffness values based on vibration
ranged from 2-31% of the measured values (α>1). Finally, a discussion of the
considerations for field application of the method is included.
Keywords timber joints, non-destructive testing, rotational stiffness, vibration
1.
INTRODUCTION
1.1. Overview
There is a growing recognition of the need for a more sustainable approach to construction. With this
awareness comes an interest in studying existing buildings as a resource for redeveloping the built
environment. Existing buildings can be deconstructed and the individual structural members reused,
however “the greenest building is the one that is already built” (Carl Elefante). Successful reuse of
entire buildings will depend on the ability to determine their current structural condition.
To be able to structurally model an existing timber frame, the member properties (strength, stiffness,
and dimensions) of the elements must be known, as well as the boundary conditions (strength and
stiffness of joints) for each member. While extensive research has been performed to nondestructively evaluate the member material properties (e.g. Ross et al, 2004, Kasal et al, 2009), the
research on in-situ determination of the member joint properties is relatively limited. In identifying
critical research needs for the American Society of Civil Engineers Structural Engineering Institute,
Ron Anthony wrote “Under design loads, seldom do wood members fail in a structure unless they are
severely deteriorated. Failures generally occur at connections. Yet we have a wealth of knowledge
about wood properties, but not the behavior of connections. Unfortunately, connections are critical in
structure performance,…,and yet we do not have a reliable means to assess their condition or
capacity.” (Anthony, 2008)
1
Paul Crovella, Department of Sustainable Construction Management and Engineering, State University of
New York – College of Environmental Science and Forestry, US, plcrovella@esf.edu
2
Dr. George Kyanka, Department of Sustainable Construction Management and Engineering, State University
of New York – College of Environmental Science and Forestry, US, ghkyanka@esf.edu
1.2. Literature Review
The determination of wood properties in-situ has been studied by many researchers. Ross and Pellerin
(1994) summarize the work done on the approaches and accuracy of assessment of basic wood
properties for in-place members. Their work shows that while “relationships between NDT
parameters and performance of wood members have been established and verified”, much of what
limits NDT methods on-site are the boundary conditions of the wood member. Much work has been
done on using low frequency vibration of wood members with known boundary conditions to detect
anomalies in stiffness (and potentially strength). As a first step in developing a structural building
health monitoring system, Cai found identification of the vibration flatwise fundamental frequency of
floor joists as an effective method (r2 = 0.68 to 0.96) to determine the static flatwise MOE (Cai et al,
2000). Soltis determined that measuring natural frequency was more effective than damping ratio at
determining stiffness loss in floor systems, and that the difference between damped and undamped
frequencies in the floors was negligible due to the low damping ratio (Soltis et al, 2000). Simply
supported floors tested by Wang (2005) showed differences of less than 3% between frequencies
measured by forced and free vibration.
Fewer researchers have worked on determining joint properties in-situ. Destructive testing of
traditional timber joints to determine stiffness and strength has been researched by Brungraber
(Brungraber, 1985) and others (Bulleit, 1999), (Schmidt, 1997), (Burnett et al, 2003). Prediction of
tensile strength for different traditional timber joints with specified geometry ratios has been adopted
by the Timber Frame Engineers Council based on work by Miller and others (2010). Anthony has
found x-ray techniques effective to reveal in-situ joint geometry and advanced decay. Morlier, Bos,
and Castéra (2006) used modal analysis to predict wood joint stiffness in a small scale wood portal
frame. However none of these methods are appropriate for rapid evaluation of existing structural
assemblies.
1.3.
Theory
The fundamental frequency for a uniform beam is governed by the beam’s geometry, material
properties, and the joint stiffness (Eq. 1). If the geometry, material properties, and natural frequency
can be determined non-destructively, the stiffness of the joints can be determined. This equation is
derived by combining expressions for the displacement of beam with the equation of motion of a
harmonic oscillator. The solution of this combined equation results in the general form sine, cosine,
and hyperbolic sine and cosine. The exact shape is found by applying boundary conditions to the
solution. This solution can then be expressed in terms of the frequency that matches the boundary
conditions. For a uniform beam, the resulting equation is:
fn = Kn (EI/ w)1/2
2πl2
(1)
Where fn = Natural frequency of mode n (cycles per second), Kn = Constant dependent on the
boundary conditions and mode, w= uniform mass per unit length (including beam mass), E = modulus
of elasticity, I = second moment of area, l= length of beam
The value of Kn for the first mode of vibration is 9.87 for a simply supported beam, and 22.4 for a
beam fixed at each end. (Figure 1)
A useful expression for describing the joint stiffness between the two extreme values is the
“normalized frequency” which represents a linear variation between 0 for a simply supported beam,
and 1.0 for a fixed-fixed beam.
Figure 1 –Fundamental Frequency of a beam depending on boundary conditions (adapted from McGuire, 1995)
If the beam properties and fundamental frequency are determined, the constant Kn can be calculated as
shown in Equation 2
Kn = fn 2πl2
(EI/ w)1/2
(2)
The value of Kn is not the value of the rotational stiffness of the joint, but rather a value that can be
correlated to the rotational stiffness of the joint. An important point arises here: When considering a
beam with semi-rigid joints, whether the joint stiffness will be significant in the analysis depends on
the ratio of the joint stiffness to the bending stiffness of the beam. As shown in Figure 2, when the
beam flexural or the joint rotational stiffness is large in relation to the other, the effect of changing
joint stiffness has a minimal effect on frequency. When the joint stiffness is between 1 and 100 times
that of the flexural stiffness, a change in joint stiffness produces a relatively large change in frequency.
1
0.9
Normalized frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.001
0.1
10
1000
100000
α = kL/EI
Figure 2 – Effect on natural frequency of varying rotational joint stiffness relative to beam stiffness
Research by Leichti and others (1995) looks at the difficulty of creating a moment resisting connection
in wood, and suggests that based on published tests, a 50% rigid connection is a possible maximum
value for wood connections due to the deformation of wood around the embedded fasteners. From
Figure 2, this corresponds to values on the baseline curve that fall below 0.5 normalized frequency.
2.
METHODOLOGY
2.1. Experimental Apparatus
The frequency was determined using an accelerometer placed on the beam to produce an electrical
signal due to the free vibration of the member. The accelerometer was a quartz shear type single-axis
accelerometer (500 mV/g). The accelerometer was attached through a 9 m (30 ft) foot cable to a USB
powered signal conditioner. This signal conditioner is powered through the USB port of the computer,
and provides a constant current input to the accelerometer. The voltage at which this current flows is
measured by the signal conditioner, and then amplified and output to the computer via a stereo jack.
The sound card of a personal computer accepted the stereo jack signal as an input. A digital audio
editor “Audacity 1.2.6" was used to capture, edit, and save the signal as a wav format file. The signal
was then processed using a Fast Fourier Transform (FFT), with code based on an applet from Evan
Mroz. The resulting frequency response spectrum output file was imported to Excel® to identify the
maximum spectral component for the first mode natural frequency.
2.2.
Beam testing
2.2.1. Validation
The first phase of the research studied the ability of frequency measurements to predict the rotational
stiffness of softwood and hardwood beams connected to a rigid test frame. The first test used a
softwood beam, Tsuga canadensis, 18.4 cm x 18.4 cm x 3.0 m (7.25 in x 7.25 in x 120 in) clamped at
each end to a continuous steel beam. The steel beam had a flexural stiffness much greater than the
wood beam, allowing the wooden member to be vibrated relative to the rigid steel frame. The center
of the wood beam was instrumented with the accelerometer and then the beam was struck by hand and
allowed to freely vibrate. The beam properties (MOE, I, density) and measured frequency were used
to calculate rotational stiffness of the joints. This was compared to the rotational stiffness as measured
by center-point loading the beam and measuring the deflection. The deflection of a beam with equal
semi-rigid joints can be determined in closed form (Pilkey, 2005), and the measured deflection was
used to determine the rotational stiffness. The experimental set-up for this work is shown in Photo 1
Photo 1 – Beam in test apparatus and data acquisition equipment
Next, a similar size beam of hardwood, Quercus rubra, was studied under the same two conditions
(simply supported and clamped). The results of frequency testing the two beams are shown in Figure
3.
1
0.9
0.8
Normalized frequency
0.7
0.6
0.5
Frequency Softwood
0.4
Frequency Hardwood
0.3
0.2
0.1
0
0.001
0.01
0.1
1
10
100
1000
10000 100000
α = kL/EI
Figure 3 – Comparison of joint stiffness measured by frequency to deflection – clamped and simply supported.
2.2.2 Beam with Varying Joint Stiffness
Testing of joints with intermediate rotational stiffness was done by using the softwood beam in the
fixture, but progressively cutting through the clamps at each end to decrease the stiffness of the
restraints (Photo 2).
Photo 2 - Image of metal plates being cut through to reduce their stiffness
1
0.9
0.8
Normalized Frequency
0.7
0.6
0.5
Frequency - softwood
0.4
0.3
0.2
0.1
0
0.001
0.01
0.1
1
10
100
1000
10000 100000
α = kL/EI
Figure 4 - Comparison of joint stiffness measured by frequency, to stiffness measured by deflection as joints are
weakened
2.3 Frame Testing
Next, two timber “H” frames were fabricated with Pinus strobus to test the effect of creating joint
stiffness using only timber joinery. The frames had identical overall dimensions: two 17.75 cm x
17.75 cm x 3.65 m posts connected to a 17.75 cm x 17.75 cm x 6.10 m beam (two 7 in x 7 in x 12 ft
posts connected to a 7 in x 7 in x 21 ft beam). One of these frames was fabricated with mortise and
matching tenons of identical size (0.000 m clearance - tight). The holes in the tenon were placed 3.2
mm (0.125 in) closer to the tenon cheeks than the holes in the mortise to the face of the mortise.
When assembled with a wood dowel the tenon cheeks were forced against the mortise by the 3.2mm
(0.125 in) difference (“Draw bored”). The other frame was fabricated with mortise and tenon joints
with a uniform 3.2 mm (0.125”) clearance between all surfaces (loose). Sample dimensions are shown
in Figure 5. The frames were fabricated using a CNC joinery machine at New Energy Works in
Farmington, NY, and tested at the State University of New York in Syracuse as shown in Photo 3.
Photo 3 – “H” frame assembled for testing in lab
Figure 5 – Sample joint dimensions (loose)
The tests compared the stiffness of the joints measured by deflection to the stiffness measured by
vibration. Four boundary conditions were applied during the tests: First the beam was set in place,
but no pins were installed (simply supported). Second, a single pin was installed at each mortise.
Third both pins were installed in each mortise. Finally, with both pins installed in each mortise, the
columns were loaded with a 910 kg (2000 lb) load.
The results for the loose and stiff frames are shown in Table 1 for frequency and in Table 2 for
stiffness. Figure 6 shows the results compared to the baseline curve.
Table 1 – Frequency Measurements (Hz) for “H” Frames
Loose Frame
Loose Frame
Tight Frame
Tight Frame
Equivalent
Deflection
Vibration
Equivalent
Deflection
Vibration
Simply
supported
6.80
6.31
6.94
6.60
1 pin installed
6.77
6.98
9.32
9.00
2 pins installed
7.28
7.36
9.54
9.58
7.40
9.71
9.73
Boundary
Conditions
2 pins w/load on No data taken
column
Table 2 – Joint Stiffness (kg-m/rad) for “H” Frames w/% error
Loose Frame
Loose Frame
Tight Frame
Tight Frame
Deflection
Equivalent
Vibration
Deflection
Equivalent
Vibration
Simply
supported
0
-2940 *
0
-760*
1 pin installed
-5.1
480 *
21,900
15,200 (-31%)
2 pins installed
2,070
3,250 *
26,400
27,200 (+3%)
3,580 *
29.700
30,200 (+2%)
Boundary
Conditions
2 pins w/load on No data taken
column
* α < 1 no % error given
1
0.9
0.8
Normalized frequency
0.7
0.6
0.5
Loose Frame
Tight Frame
0.4
0.3
0.2
0.1
0
0.001
0.01
0.1
1
10
100
1000
10000 100000
α = kL/EI
Figure 6 – Stiffness of loose and tight H frames with various boundary conditions.
3.
DISCUSSION
3.1 Effectiveness of method as a function of α
The three sets of tests share common characteristics: There is a much higher degree of precision
when α > 1. In the case of the frames, this method predicts actual joint stiffness within 2-31% for α
>1. Although this range is large, for structural analysis the stiffness can vary over orders of
magnitude. This degree of precision will allow for a known degree of certainty during analysis. For
the samples with α < 1, the ability of the method to predict the stiffness is inadequate. However when
analyzing a structure, knowing that the joint stiffness is low in relation to the member stiffness will
allow the designer to assume no joint stiffness, and use a pinned model. This is an acceptable
assumption for this case.
Further, the values for the frame joints tested with double pins and with a single pin indicate that the
method can distinguish the decrease in stiffness due to the missing/failed pin (e.g. damaged joint) at
higher stiffnesses (18% strength loss vs. 12% error).
Contrary to popular opinion, a timber joint can be created with sufficient stiffness to have a significant
effect on beam stresses due to bending.
3.2 Challenges for extending work to the field.
A number of factors complicate the application of the method under field conditions: Determination of
MOE and density, assumption of equal stiffness at each joint, attachment of materials and braces, and
vibration of other parts of the structure are to be considered.
3.2.1 Determination of MOE and Density
The determination of MOE (and density) might be performed in two ways in a field scenario: A visual
grade assigned, or a small wood sample taken and tested. In the case of the H frame material,
identification of the species and grade (Eastern White Pine, Grade 2) yields an MOE= 6200 MPa
(900,000 psi) (AFPA, 2005). The increment bore samples yielded a specific gravity of 0.33. Using
the specific gravity to determine the MOE as described in the Wood Handbook (FPL, 2010), the clear
sample MOE would be 6800 MPa (990,000 psi). The ASTM procedure for correcting the value for
defects (ASTM, D245) would reduce this to 80% of the clear sample value, 5400 MPa (790,000 psi).
The Wood Handbook lists 22% as the average coefficient of variation of MOE and 10% as the average
COV of specific gravity for all species. Table 3 shows the percent change in stiffness for MOE values
calculated with various methods.
Table 3 – Values of stiffness using different methods for determining MOE
MOE Value
(MPa)
Stiffness (kg-m/rad)
Deflection
5,700
30,200
0
Species and visual grade
EWP timber, #2
6,200
24,200
20%
Measure density and ASTM
245-00
5,400
33,300
10%
4,500 -7,000
23,60036,500
21%
MOE Method
MOE of 22% COV, density
by 10% COV
Stiffness
% error
Theoretically, a transverse vibration of the timber could also yield an MOE if the joints had no
stiffness in the transverse direction.
An alternative to using the increment bore to determine the density, would be the approach used by
Minster, et al (2006) for MOE determination based on bore testing.
3.2.2 Assumption of equal stiffness at each joint –Machine fabrication of the joints ensured that the
stiffnesses of each end of the beam were similar. However, multiple researchers have modeled the
frequency response for a beam with unequal stiffness joints at each end, and the mode shape and
frequency can be calculated for these conditions. In an example by Gorman (1975), a beam with a
normalized frequency of 0.22, and individual spring stiffness of α=2.678 and α=1.53, (one spring 73%
stiffer than the other) the natural frequency of vibration would be equivalent to that of a beam with
equal springs of α = 1.85 (a 14% error compared to the arithmetic average α stiffness).
3.2.3 Attachment of materials and braces – The vibration characteristics of beams with point loads and
distributed mass have also been modeled by researchers. Vibration characteristics for continuous
beams pinned at multiple points (such as connections due to knee braces) have also been determined.
However the multiple span beam does not allow for determining joint stiffness with a single vibration
measurement. This will require modal analysis.
3.2.4 Vibration of other parts of the structure – As joints stiffness increases, more of the deformation
is transferred through the joint to the surrounding structure. As Leichti (1999) notes, typically the
ratio of column to beam dimensions is such that the columns are stiffer than the beams. Depending on
the relative stiffness of these members, their deformations can influence the vibration of the adjacent
member. While this effect may require modal analysis to determine the final behavior, a sensitivity
analysis of the effect of loading the columns of the H frame was performed. The frequency of the
beam vibration and the joint stiffness was measured as a load was placed on the column. The load was
increased in increments of 227 kg (500 pounds) from ~0 to 910 kg (2000 lbs). For the tight frame, this
caused the joint stiffness to increase by 12%.
4.
CONCLUSIONS
The understanding of the effect of joint stiffness on beam behavior is based on the relative stiffness of
the joint and beam. The vibration method does not effectively distinguish low joint stiffnesses (α < 1),
but by determining the stiffness to be low, allows the designer to use an appropriate model (pinned
joints). In the case of semi-rigid behavior (α >1), the method predicted the stiffness within 12%, on
average. Since the stiffness can vary be an order of magnitude in this range, this 12% error can be
considered in that context.
The method was able to distinguish the missing peg (simulating failure) from the double peg joint
(18% loss vs. 12% error). However much more testing would be needed to develop a clear
understanding of the average error for the α < 1 range.
A number of confounding factors for application in the field will affect the results. The method can
not distinguish when joints have unequal stiffness at each end of the beam. The method is affected by
the stiffness and load of the members supporting the beam. The method will apply only when attached
members and materials can be idealized as loads and restraints. The method will require a field
determination of the MOE. As shown in section 3.2, each of the above mentioned factors could be
expected to introduce an error of approxiamtely 10-20%.
ACKNOWLEDGMENTS
Research was supported by donations in-kind from Northcott Wood Turning and New Energy Works.
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