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Reduced and test-data correlated FE-models of a large timber truss with dowel-type connections aimed for dynamic analyses at serviceability level

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The rise of wood buildings in the skylines of cities forces structural dynamic and timber experts to team up to solve one of the new civil-engineering challenges, namely comfort at the higher levels, in light weight buildings, with respect to wind-induced vibrations. Large laminated timber structures with mechanical joints are exposed to turbulent horizontal excitation with most of the wind energy blowing around the lowest resonance frequencies of 50 to 150 m tall buildings. Good knowledge of the spatial distribution of mass, stiffness and damping is needed to predict and mitigate the sway in lighter, flexible buildings. This paper presents vibration tests and reductions of a detailed FE-model of a truss with dowel-type connections leading to models that will be useful for structural engineers. The models also enable further investigations about the parameters of the slotted-in steel plates and dowels connections governing the dynamical response of timber trusses.
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Engineering Structures 260 (2022) 114208
Available online 20 April 2022
0141-0296/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Reduced and test-data correlated FE-models of a large timber truss with
dowel-type connections aimed for dynamic analyses at serviceability level
Pierre Landel
a
,
b
,
*
, Andreas Linderholt
b
a
RISE Research Institutes of Sweden, Wood Building Technology, 501 15 Borås, Sweden
b
Linnaeus University, Department of Mechanical Engineering, 351 95 V¨
axj¨
o, Sweden
ARTICLE INFO
Keywords:
Tall timber structure
Glulam truss
Mechanical connection
Dowel-type fastener
Wind-induced vibration
Modal testing
Dynamic properties
Connection stiffness
Serviceability
FE-model reduction
ABSTRACT
The rise of wood buildings in the skylines of cities forces structural dynamic and timber experts to team up to
solve one of the new civil-engineering challenges, namely comfort at the higher levels, in light weight buildings,
with respect to wind-induced vibrations. Large laminated timber structures with mechanical joints are exposed to
turbulent horizontal excitation with most of the wind energy blowing around the lowest resonance frequencies of
50 to 150 m tall buildings. Good knowledge of the spatial distribution of mass, stiffness and damping is needed to
predict and mitigate the sway in lighter, exible buildings. This paper presents vibration tests and reductions of a
detailed FE-model of a truss with dowel-type connections leading to models that will be useful for structural
engineers. The models also enable further investigations about the parameters of the slotted-in steel plates and
dowels connections governing the dynamical response of timber trusses.
1. Introduction
Perceptible vibrations in lightweight structures are often annoying to
inhabitants and hard to mitigate. For instance, vertical vibrations
induced by footsteps in timber oors have been investigated for a long
time [1,2], and now, models predicting this behavior and design rules to
achieve a good comfort are available in standards and building codes.
Meanwhile, as trendy timber buildings reached 85 m in height or grown
up to 20 stories [3], the horizontal sway induced by the wind has
become an important and new application for structural dynamics. At
serviceability level, the wind-induced vibrations (in both along-wind
and across-wind directions) have been highlighted as key problems in
taller multi-story wooden buildings [39]. To date, only a few models
that correctly predict the response of tall timber structures for habit-
ability have been proposed [1013]. Experimental studies focusing on
the dynamic responses of timber structures at different scales and
serviceability levels have produced valuable data for improving the
accuracy of these models. Indeed, vibration and cyclic tests have been
performed to explore stiffness and energy dissipation in engineered
timber products [1418], in mechanical connections with steel plates
and dowels [19,20], and for different lateral-resisting truss or frame
prototypes [2124]. During the last decade, many ambient vibration
campaigns, on real buildings, that focus on the lowest eigenfrequencies
and damping ratio have been performed [2528], and recently, taller
timber buildings have been excited with mass inertia shakers to improve
estimates of modal properties [2932].
Timber trusses used in bridges or buildings are usually modelled with
beam elements and either pinned or clamped joints, but consideration of
translational stiffness at the joints is recommended for better prediction
of dynamical response [7,24] and for better load distribution at the
joints [33], whereas rotational stiffness should be considered for frames
with semi-rigid connections between columns and beams [34]. Other
local parameters of interest are the clearance around the dowels and the
manufacturing tolerances that lead to an initial slip and thus a nonlinear
response according to [20,3537]. Many dowel-joint modelling ap-
proaches have been analyzed in static and these can be useful for dy-
namic prediction of comfort in tall timber buildings, although they often
focus on survivability issues such as load carrying capacity, ductility, or
fatigue strength. The European design code for timber structures,
Eurocode 5 [33], recommends a simple equation for the slip modulus,
K
ser
, of timber-dowel shear-connection for serviceability; however, the
optimal value of K
ser
for any given application is uncertain. From pre-
vious experimental tests, K
ser
was found to underestimate the joint
stiffness by a factor of about two [20,23,35] and in other test K
ser
was
* Corresponding author at: RISE Research Institutes of Sweden, Wood Building Technology, 501 15 Borås, Sweden.
E-mail address: pierre.landel@ri.se (P. Landel).
Contents lists available at ScienceDirect
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
https://doi.org/10.1016/j.engstruct.2022.114208
Received 4 August 2021; Received in revised form 21 February 2022; Accepted 27 March 2022
Engineering Structures 260 (2022) 114208
2
found to sometimes overestimate the stiffness by a factor of about 1.7
[7].
To address some of these issues with vibration and timber structures,
this paper explores the dynamical properties of a large glued laminated
(glulam) truss with slotted-in steel plates and dowels connections that is
used as lateral bracing structure in a tall timber building. First, the
experimental results from vibration tests on the truss and on its diverse
components will be presented. Next, detailed FE-models of the truss will
be explained, and results from modal analysis will be correlated to the
test data. Finally, reduced models of the truss will be described, and the
speed of analysis and the accuracy of the modal results will be discussed.
An analogy with a simple model developed earlier [24] using connection
stiffness stemming from the Eurocode 5 will be included. The paper will
conclude with a discussion of the model reductions and the various
comparisons, and of forthcoming investigations of this large glulam
truss with slotted-in steel plates and dowels connections.
2. Experimental investigations
2.1. The truss
The large glulam truss investigated in this paper was manufactured
and assembled by the company Moelven T¨
oreboda and is one of the
structural components in an eight-story residential building in Sk¨
ovde,
Sweden. The 18.5 m tall and 4.3 m wide truss stands vertically over six
stories and contributes to the lateral bracing (Fig. 1). The truss is planar
and consists of fourteen glulam members of quality GL30c according to
[38] with a width of 215 mm and heights of 360 and 540 mm. The
members have rectangular cross-sections except for the left column
which has a T-shaped cross-section. The truss is assembled with two 8
mm thick slotted-in steel plates at each of the 14 connections and with
650 dowels of diameter 12 mm (see the production drawing and details
in Figs. 2 and 3). The dowels were smooth, but some had threaded ends
with washers and nuts to secure transport and lift of the assembled truss.
Both the steel plates and the dowels were made of S355JO steel quality.
After CNC cutting, the glulam and steel elements were weighed, and
the lengths and the cross-sectional dimensions of the members were
measured. The densities
ρ
were calculated from the measured weights
and the net volume of the beams (considering the holes and the slots)
and results are reported in Table 1 together with the dimensions. The
moisture contents of the members were recorded at different locations
and the mean values per member varied between 12.5% and 15.6%.
The total weight of the truss was 4,284 kg with 3,460 kg for the
glulam and 824 kg for the steel parts (plates and dowels).
Glulam members of quality GL30c can, according to [38], have lay-
ups combining 45 mm thick lamellas of different qualities in specic
proportion and places. For the short and smaller beams, the manufac-
turer used two lamellas of quality T22 for the outer zones and four la-
mellas of lower quality T15 for the inner zone of the beam. The columns
pc80 and pc81 were produced with three outer lamellas and six inner
lamellas of qualities T22 and T15, respectively. The number in the T-
classication refers to the fth percentile tensile strength of the board
and requirements for stiffness and density of the lamellas and GL30c
glulam are summarized in Table 2.
Assuming normal distributions for E
x
and
ρ
for GL30c glulam, the
variation coefcient for the modulus of elasticity E
x
of GL30c glulam is
about 10% and for the density it is 6% (both at a condence level of
75%) in the standard.
2.2. Modal tests on four of the glulam members
Impact testing with an impulse hammer and three single-axis ac-
celerometers was performed on three beams (b107-2, b107-3 and b107-
4) and on the column pc80. The members were already CNC-cut with 10
mm wide slots and ø 12 mm holes at both ends. The short-sledge impulse
hammer (model 086D20 from PCB Piezotronics) had a sensitivity of
0.23 mV/N (±15%) and the ceramic shear accelerometers (model
T333B30 from PCB Piezotronics) had a sensitivity of 100 mV/g (±10%).
The glulam members hung at their center of gravity from an overhead
crane. The impulse and the acceleration were measured at one end of the
structural element. Only the response in one DOF was recorded at a time,
that is, in the direction of the impact, either axially (in the local x-di-
rection) or transversally (in the local y- or z-direction) (Fig. 4).
The data were analyzed with the software Signal Calc ACE from
DataPhysics to estimate the natural frequencies of the different modes
(Table 3). During the tests, the ambient temperature was 2021 C and
Fig. 1. (a) 3D-view from the CAD-Model of the building structure with the truss studied in blue, (b) picture from the building site with the truss clamped to the
concrete podium below.
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
3
the relative humidity 4045%.
Both axial and transversal modes were easier to identify for the
relatively slender column pc80. The eigenfrequencies of the two rst
axial modes of the three shorter beams with the same geometrical di-
mensions were similar even though mass densities varied (Table 1). The
transversal modes were more difcult to measure on the shorter beams
and for b107-3 and b107-4, they were not accurately identied. The
impact from the support condition (hanging at the center of gravity) on
the vibration modes have not been assessed in this study. Although,
hanging of timber products has been used to experimentally identify the
modal characteristics [15,18] and studies on different support condi-
tions have highlighted a relatively low impact on the frequencies of
bending modes (lower than 2%) [40].
2.3. Vibration tests of the large glulam truss
The truss was assembled in the factory by situating the glulam
members, adding the steel plates in the 10 mm slots, and inserting the
dowels into the ø 12 mm holes. The forced vibration tests were also
performed in the factory. The truss was lifted from the ground with an
overhead crane using straps placed close to the center of gravity of the
truss. With the short-sledge impulse hammer, excitations were induced
in two directions at the steel foot xed to the end of the column pc80.
Fig. 2. Exploded 3D-view of (a) a T-connection and (b) a KT-connection.
Fig. 3. Production drawing of the assembled large glulam truss with beam
(b10xx), column (pcxx) and connection (Cxx) identication numbers.
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
4
Fourteen tri-axial piezo electrical accelerometers at the center of each
truss connection and twelve single-axis accelerometers at the middle of
the short glulam elements were used to measure motion (Fig. 5). All
accelerometers were glued to the same side of the truss.
The data were recorded with an LMS data acquisition system with 56
input channels measuring the excitation force from the impulse hammer
and the accelerations at 50 Degrees of Freedom (DOFs) in the XYZ-
coordinate system of the truss. In the residential building, the Z-axis is
parallel to the gravitational load and the Y-axis correspond to one wind-
load direction. Thus, the YZ-plane is the main working plane of the truss.
The recorded accelerations and the excitation force were analyzed with
the software LMS Test.Lab to extract Frequency Response Functions
(FRFs) for each impulse test. During the tests, the ambient temperature
was 20 C and the relative humidity 30%. From the averaged data in the
frequency domain, modal analysis was used to identify eigenmodes (i.e.,
eigenfrequencies, damping values, modal masses and mode shapes).
When excited in the Y-direction, ve eigenmodes with eigenfrequencies
between 9 and 93 Hz with motions mainly in the plane of the truss (i.e.,
the YZ-plane) were identied. The data from the 38 DOFs at the 14
connections were used (Table 4). When excited in the X-direction, 62
eigenmodes with motion in 3D and eigenfrequencies between 1 and 102
Hz were identied using the data from all the 50 DOFs (Table 5). The
auto-MAC (Modal Assurance Criterion) of the measured eigenmodes
showed good orthogonality with values below 0.03 for two different
modes.
The eigenfrequencies of the in-plane bending modes of the truss
identied here (i.e., 42.6, 61.6 and 92.8 Hz) were much higher than the
lowest eigenfrequencies of tall timber buildings. Here, the aim is to make
models that are valid in their representation of the real structure. When
the truss is connected to other components it renders in low-frequency
modes. Indeed, the rst bending frequencies of six- to eighteen-storey
timber buildings have been measured or calculated to be between
0.32 and 3.8 Hz [8,26,27]. Meanwhile, dynamical studies on single
structural parts bring relevant knowledge for prediction of the
Table 1
Measured dimensions and densities for the glulam members.
Component
ID
Cross-section and dimensions*
(mm)
Length
(mm)
Density (kg/
m
3
)
b102 R 214 ×359 4,110 441.2
b103 R 214 ×359 4,222 420.1
b104 R 214 ×359 4,223 426.0
b105 R 214 ×359 4,223 455.7
b106 R 214 ×359 4,223 438.3
b107-1 R 214 ×359 3,226 428.7
b107-2 R 214 ×359 3,226 413.6
b107-3 R 214 ×359 3,226 415.8
b107-4 R 214 ×359 3,226 433.9
b107-5 R 214 ×359 3,226 434.0
b107-6 R 214 ×359 3,226 426.5
b108 R 214 ×359 4,246 406.2
pc80 R 214 ×539.5 18,245 434.8
pc81-5253 T 391 ×538.5 18,245 425.4
Mean 428.6
CoV 2.8%
*R for rectangular cross-sections and T for T-shaped cross sections.
Table 2
Stiffness parallel to the grain and density values of T15- and T22-classied lamellas according to [39] and glulam members classied as GL30c according to [38].
Mean longitudinal modulus of elasticity in tension
E
mean
Lowest fth percentile longitudinal modulus of elasticity in
tension E
x,05
mean density
ρ
mean
fth percentile density
ρ
05
T15 11,500 MPa 7,700 MPa 430 kg/m
3
360 kg/m
3
T22 13,000 MPa 8,700 MPa 470 kg/m
3
390 kg/m
3
GL30c 13,000 MPa 10,800 MPa 430 kg/m
3
390 kg/m
3
Fig. 4. Test setup with the hanging beam, the accelerometers, and the beams local xyz-coordinate system.
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
5
dynamical response of a complete building and at least the lowest two or
three global bending modes of the sub-parts should be investigated. The
spatial distributions of the stiffness and the mass govern the modal
properties, such as the eigenmodes of a sub-structure, and can be studied
independently. Moreover, the vibration amplitudes in terms of
displacement and acceleration during an FVT should be relevant as the
stiffness and the damping properties of timber structure might be
nonlinear, i.e., they depend on the level of vibration. The maximal ac-
celeration, that is the rms-value, during the test is estimated to 400 mm/
s
2
at 119 Hz. The maximum rms-acceleration, during the tests, at the
second in-plane mode, at 25 Hz, is estimated to 130 mm/s
2
. The rms-
values are estimated from backward calculations setting out from
power spectral densities. Comparatively, 40 mm/s
2
is the horizontal
peak acceleration limit for a residential building with a rst natural
frequency of 1 Hz [41].
3. FE-Models for modal analyses and reductions
3.1. Solid elements FE-models
A large, detailed 3D FE-model of the glulam truss with steel plates
and dowels, (here called model D
1
), was developed with MSC SimXpert
and Matlab and analyzed with MSC Nastran. According to the produc-
tion drawings, a 3D-CAD model was drawn with dimensions adjusted
from the measurements and imported in MSC SimXpert to mesh the
glulam and the steel parts with eight- and six-noded solid elements.
Different element sizes have been used with densied meshes around
details such as the holes for the dowels in the timber members and the
steel parts and coarser meshes in areas with fewer geometrical in-
homogeneities (Fig. 6). To ensure continuity between solid elements
from the same part but with non-congruent meshes, contact constraints
paired and permanently glued slave to master bodies. Permanent glue
contacts also constrained parts of different materials together: the tim-
ber members to the ø 12 mm dowels and the dowels to the steel plates.
The 0.25 mm gaps between the dowels and the plates, bored with 12.5
mm holes, were kept constant, i.e., the glue property acted as a ller,
and friction, penetration and separation between glued parts were
disabled. One-millimeter gaps, without any contacts, between the tim-
ber and the steel plates were modelled. During the experiment, the two
parts might have been in contact with each other, but the effect on the
in-plane eigenmodes is assumed to be small. Model D
1
comprised
1,726,778 eight-noded solid elements (CHEXA) and 523,961 six-noded
solid elements (CPENTA) with a total of about 8,147,000 DOFs, or
2,715,641 nodes. The glulam was modelled as an almost transversely
Fig. 5. Setup of the vibration test with the truss hanging in Moelvens factory.
Table 3
Eigenfrequencies of the measured vibration modes of four beams.
Experimental modes b107-2 in Hz b107-3 in Hz b107-4 in Hz pc80 in Hz
Axial mode 1 f
1,x
=794.0 f
1,x
=805.8 f
1,x
=797.5 f
1,x
=138.7
Axial mode 2 f
2,x
=1,592.5 f
2,x
=1,589.3 f
2,x
=1,579.5 f
2,x
=276.2
Axial mode 3 f
3,x
=424.1
Axial mode 4 f
4,x
=557.2
Axial mode 5 f
5,x
=692.0
Transv. mode 1 f
1,y
=104.7 f
1,z
=8.6
Transv. mode 2 f
2,y
=250.8 f
2,z
=23.1
Transv. mode 3 f
3,y
=402.4 f
3,z
=43.7
Transv. mode 4 f
4,y
=453.2 f
4,z
=69.3
Transv. mode 5 f
5,z
=98.4
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
6
isotropic elastic material with six stiffness parameters determined ac-
cording to the next section of the paper and the measured densities. The
number of independent constants is reduced to ve in transverse isot-
ropy, but in the case of timber the relationships between the elasticity
moduli, the shear moduli and the Poissons ratio are not obvious and six
independent constants yield an appropriate model for this study. The
steel was modelled as an isotropic elastic material with a Youngs
modulus of 210 GPa, a Poissons ratio of 0.3 and a density of 7,850 kg/
m
3
. Two spring elements acting in the Y-direction and representing the
hoisting loops were attached to the ground at one end of the spring
element, and to several points of the upper timber column with an
interpolation constraint element (RBE3 in MSC Nastran) at the other end
of the spring element. The RBE3 elements dened the motion at a
reference grid point as the weighted average of the motions at a set of
other grid points. The same axial stiffness value for the two springs was
set to match the natural frequency of the global bouncing mode in the Y-
direction as that found from the earlier experimental results.
3.2. Glulam material calibration in four steps
3.2.1. Step 1: Stiffness values according to axial vibration and standard
values
Among the six independent stiffness parameters in the material
model used for the glulam parts, each beams longitudinal modulus of
elasticity, Ex, was rst estimated from the mean of the eigenfrequencies
of the lowest axial modes identied from the impact tests (see Section
2.2). Assuming an Euler-Bernoulli beam of length L, density
ρ
, and a
homogeneous and isotropic material, several En,x were estimated from
the eigenfrequencies fn,x of the axial vibration modes (parallel to the
beams x-direction), according to the analytical equation (1) and they
are presented in Table 6 together with the eigenfrequencies of the
measured axial modes.
En,x=
ρ
2L fn,x
n2
with nbeing the mode number (1)
The shear moduli Gxy estimated from the rst transversal eigenfre-
quency of the beam b107-2 by assuming Timoshenko beam theory with
Bordonn´
es derivation [14] is 1,570 MPa but no realistic value yields
from the rst transversal eigenfrequency of the beam pc80. The eigen-
frequencies of the second bending mode yield shear moduli values of
580 MPa and 3,100 MPa respectively for the beams b107-2 and pc80.
Therefore, standard values for glulam of GL30c quality according to [38]
were adopted for the following stiffness parameters of the glulam ma-
terial model: Gxy =Gzx =650 MPa, Ey=Ez=300 MPa and Gyz =
Gzy =65 MPa. The Poissons ratios have been taken from [42] with
υ
xy =
υ
xz =0.5;
υ
yz =
υ
zy =0.2. The material stiffness properties were
entered in the FE-model of the beams using a three-dimensional aniso-
tropic material model (MSC Nastran MAT9 card).
3.2.2. Step 2: Sensitivity screening on the FE-model
A minor sensitivity analysis on the stiffness parameters was per-
formed to understand their impact on the lowest elastic eigen-
frequencies. This analysis was made using an FE-model of a 10 m long
rectangular prism with cross-section dimensions of 214 ×540 mm, thus
without slots or holes, and with the material dened in the previous
section. From the ten lowest numerical eigenmodes, the following ob-
servations were made:
- Only torsional and transversal, no axial, eigenmodes,
- Of course, Ex impacted the transversal modes a lot (about 9% higher
eigenfrequencies for 20% higher stiffness), but not the torsional
modes,
- Gyz =Gzy had little impacts on the eigenfrequencies (about 1% for
20% change of stiffness),
Table 4
The ve lowest experimental in-plane dominated mode shapes when excited in the Y-direction. The DOFs at the middle of the diagonals and the short beams are
calculated as the average displacements in the plane at the end of these members.
Mode number and description Eigenfrequency (relative viscous damping) Mode shape visualization
1. Bouncing in the Y-direction 9.0 Hz
(1.7%)
2. Column end moving in the Y-direction 25.2 Hz
(0.9%)
3. First global bending mode around the X-axis 42.6 Hz
(1.0%)
4. Second global bending mode around the X-axis 61.6 Hz
(0.6%)
5. Third global bending mode around the X-axis 92.8 Hz
(0.7%)
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
7
- Gxy =Gzx had little impacts on higher bending modes (about 1% for
20% change of stiffness) and large impacts on the torsional modes
(about 9% for 20% higher stiffness),
- The parameters Ey=Ez,
υ
xy =
υ
xz and
υ
yz =
υ
zy had no or very little
impact (less than 1% for 20% stiffness difference) on the ten lowest
eigenfrequencies.
3.2.3. Step 3: Stiffness calibration of the FE-models
The outcome of the sensitivity screening was that the longitudinal
modulus of elasticity was the main important stiffness parameter to
calibrate in the FE-models. Therefore, in the third step, only the modulus
of elasticity was in focus. Using the mean modulus of elasticity obtained
in the rst step and presented in Table 6, for each of the FE-models of the
Table 5
Some of the most visually interesting experimental modes when excited in the X-direction.
Mode number and description Eigenfrequency (relative viscous damping) Mode shape visualization
2. First global twist/torsion around the Z-axis 2.3 Hz
(1.9%)
3. First global bending around the Y-axis 2.7 Hz
(0.6%)
4. Second global twist/torsion around the Z-axis 5.4 Hz
(1.4%)
5. Second global bending around the Y-axis 6.4 Hz
(0.7%)
8. Third global bending around the Y-axis 13.5 Hz
(0.6%)
10. Fourth global bending around the Y-axis 18.9 Hz
(0.8%)
Fig. 6. Detailed view of the mesh with solid elements of a KT-connection in model D
1
, the yellow lines represent the surfaces of the solid elements with permanently
glued contact.
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
8
four beams, the eigenfrequencies of the axial modes were calculated
with MSC Nastran. Since the relationship between eigenfrequencies and
longitudinal modulus of elasticity follows the power of two propor-
tionality according to equation (1), then Ex can be updated according to
equation (2) using the mean factor kf. The factor and the updated Ex,num2
are presented in Table 7. The values of kf were small and indicated that
the Euler-Bernoulli theory yields good estimation of the axial eigen-
frequencies of beams with non-uniform geometries and mass
distribution.
Ex,num2=kfEx,num1with kf
=
n
i=1
1
nfi,x,exp
fi,x,num12
and nbeing the number of modes (2)
The eigenfrequencies of the axial and transversal modes of the beams
with both values of Ex and with the standard value for GL30c
(Ex,std =13 GPa and
ρ
=430 kg/m3) are presented in Table 8 and
compared to the experimental results. The transversal eigenfrequencies
were well estimated for both calibrated models (Ex,num1 and Ex,num2) with
errors up to 3%. Both axial and transversal eigenfrequencies were
overestimated between 3 and 9% with the standard values of Ex and
ρ
for
pc80 and b107-2. Fig. 7 shows the third numerical transversal mode of
element b107-2.
The accelerometer used to measure transversal vibrations was placed
some decimeters from the end of the beam to avoid the area with the
slots. The small displacements at some places (e.g., close to nodal points)
could make it difcult to identify the higher transversal modes and could
also be the reason that the transversal modes of the short beams were not
easy to measure.
Table 6
Mean values of measured eigenfrequencies (in Hz) in the axial direction and corresponding Euler-Bernoulli evaluations of the longitudinal modulus of elasticity (in
MPa) for the four beams tested.
Beam pc80 Beam b107-2 Beam b107-3 Beam b107-4
Axial vibration
mode
Exp.
eigenfreq.
Evaluated modulus
of elasticity
Exp.
eigenfreq.
Evaluated modulus
of elasticity
Exp.
eigenfreq.
Evaluated modulus
of elasticity
Exp.
eigenfreq.
Evaluated modulus
of elasticity
Mode 1 f
1,x
=138.5 E
1,x
=11,146 f
1,x
=794.0 E
1,x
=10,862 f
1,x
=805.8 E
1,x
=11,239 f
1,x
=797.5 E
1,x
=11,488
Mode 2 f
2,x
=276.2 E
2,x
=11,046 f
2,x
=
1,592.5
E
2,x
=10,923 f
2,x
=
1,589.3
E
2,x
=10,929 f
2,x
=
1,579.5
E
2,x
=11,265
Mode 3 f
3,x
=424.1 E
3,x
=11,569
Mode 4 f
4,x
=557.2 E
4,x
=11,234
Mode 5 f
5,x
=692.0 E
5,x
=11,088
Mean E
x,pc80
¼11,216 E
x,b107-2
¼10,893 E
x,b107-3
¼11,084 E
x,b107-4
¼11,376
Table 7
Update of the longitudinal modulus of elasticity for the numerical model.
Components Ex,num1 in MPa kf Ex,num2 in MPa
pc80 11,216 1.006 11,290
b107-2 10,893 0.993 10,820
b107-3 11,084 0.995 11,030
b107-4 11,376 0.996 11,325
Table 8
Experimental and numerical eigenfrequencies (in Hz) of the four tested components with different values of Ex and percentage differences in brackets.
Table 8a Component pc80
Vibration mode Exp. eigen-frequency Num. freq. 1
Ex =11 216 MPa
ρ
=435 kg/m
3
Num. freq. 2
Ex =11 290 MPa
ρ
=435 kg/m
3
Num. freq. std.
Ex =13 000 MPa
ρ
=430 kg/m
3
Axial mode 1 f
1,x
=138.7 139.1 (+0.2%) 139.5 (+0.5%) 150.5 (+8.5%)
Axial mode 2 f
2,x
=276.2 277.4 (+0.4%) 278.2 (+0.7%) 301.1 (+9%)
Axial mode 3 f
3,x
=424.1 420.5 (0.8%) 421.8 (0.5%) 455.0 (+7.3%)
Axial mode 4 f
4,x
=557.2 554.5 (0.5%) 556.3 (0.2%) 599.9 (+7.6%)
Axial mode 5 f
5,x
=692.0 685.9 (0.9%) 687.9 (0.6%) 733.0 (+5.9%)
Transv. mode 1 f
1,z
=8.6 8.4 (2.9%) 8.4 (2.6%) 9.1 (+4.9%)
Transv. mode 2 f
2,z
=23.1 22.6 (2.1%) 22.7 (1.8%) 24.3 (+5.4%)
Transv. mode 3 f
3,z
=43.7 42.7 (2.5%) 42.8 (2.2%) 45.7 (+4.5%)
Transv. mode 4 f
4,z
=69.3 67.3 (2.9%) 67.5 (2.7%) 71.7 (+3.4%)
Transv. mode 5 f
5,z
=98.4 95.5 (2.9%) 95.7 (2.7%) 101.2 (+2.9%)
Table 8b Component b107-2
Vibration mode Exp. eigen-frequency Num. freq. 1
Ex =10 893 MPa
ρ
=414 kg/m
3
Num. freq. 2
Ex =10 820 MPa
ρ
=414 kg/m
3
Num. freq. std.
Ex =13 000 MPa
ρ
=430 kg/m
3
Axial mode 1 f
1,x
=794 802.4 (+1.1%) 799.84 (+0.7%) 859.54 (+8.3%)
Axial mode 2 f
2,x
=1 592.5 1 586.3 (0.4%) 1 581.2 (0.7%) 1694.7 (+6.4%)
Transv. mode 1 f
1,y
=104.7 104.5 (0.2%) 104.22 (0.5%) 111.14 (+6.1%)
Transv. mode 2 f
2,y
=250.8 255.1 (+1.7%) 254.5 (+1.5%) 266.97 (+6.4%)
Transv. mode 3 f
3,y
=402.4 412.2 (+2.4%) 411.4 (+2.2%) 425.67 (+5.8%)
Table 8c Component b107-3 Component b107-4
Axial mode Exp. eigen-frequency Num. freq. 1
Ex =11 084 MPa
ρ
=416 kg/m
3
Num. freq. 2
Ex =11 030 MPa
ρ
=416 kg/m
3
Exp. eigen-frequency Num. freq. 1
Ex =11 376 MPa
ρ
=434 kg/m
3
Num. freq. 2
Ex =11 325 MPa
ρ
=434 kg/m
3
Mode 1 f
1,x
=805.83 807.48 (+0.2%) 805.52 (+0.0%) f
1,x
=797.54 800.99 (+0.4%) 799.1 (+0.2%)
Mode 2 f
2,x
=1 589.3 1 593.8 (+0.3%) 1 590.90 (+0.1%) f
2,x
=1 579.5 1 580.20 (0.04%) 1 577.3 (0.1%)
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
9
3.2.4. Step 4: Linear regression for the other glulam members
The Ex values of the four members calibrated in the previous step
showed a mostly linear proportionality with their densities (Fig. 8), with
a coefcient of determination of 0.9.
Because the other ten glulam members could not be calibrated from
the modal experiment due to either complex geometry or bad vibration
test results, their Ex values (Table 9) were estimated using the equation
describing the linear regression (Fig. 8) and the densities (Table 1).
3.3. Comparative analysis of the numerical and experimental modal
results
Modal analysis (SOL103 in MSC Nastran) was used to identify the
eigenmodes with eigenfrequencies below 120 Hz for the assembled and
spring supported glulam truss model (model D
1
) described in the pre-
vious section. The analysis resulted in 95 eigenmodes, see two of them in
Fig. 9. An AutoMAC matrix was calculated, and the mode shapes showed
good orthogonality to each other, with most of the off-diagonal values
close to zero. Only a few AutoMAC values of different eigenmodes were
between 0.4 and 0.55.
Using MAC values, the numerical mode shapes were compared at the
same DOFs as the experimental mode shapes obtained with excitation in
the Y-direction and in the X-direction. The eigenfrequencies of the
paired experimental and numerical eigenmodes with MAC values higher
than 0.85 are presented in Tables 10 and 11. The same analysis was
performed using another detailed FE-model, here called D
2
, with stan-
dard values for density (430 kg/m
3
) and longitudinal modulus of
Fig. 7. The third numerical transversal mode (at 411.4 Hz) of b107-2 with
rened material values.
Fig. 8. Linear regression between density and E
x for the four beams tested.
Table 9
Longitudinal modulus of elasticity for the glulam mem-
bers in the detailed FE-model D
1
with bold text for the four
members calibrated in step 3.
Glulam members ID Ex in MPa
b102 11,450
b103 11,030
b104 11,150
b105 11,740
b106 11,390
b107-1 11,200
b107-2 10,820
b107-3 11,030
b107-4 11,325
b107-5 11,310
b107-6 11,160
b108 10,750
pc80 11,290
pc81-5253 11,130
Mean 11,200
CoV 2.2%
Fig. 9. Numerical mode shapes of model D
1
: (a) the second global bending
mode out-of-plane, (b) the rst global bending mode in-plane.
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
10
elasticity (13 GPa) for the glulam members. The results from the com-
parison are also presented in Tables 10 and 11.
The eigenmodes from the detailed FE-model D
1
matched well with
many of the experimental eigenmodes in terms of mode shapes and
eigenfrequencies. All ve experimental in-plane eigenmodes were well
predicted by the model D
1
. This model also well predicted the out-of-
plane eigenmodes with eigenfrequencies below 32 Hz, whereas it did
not perform well at matching the out-of-plane eigenmodes with higher
eigenfrequencies. Generally, the matched numerical eigenmodes, in
terms of mode shapes, showed slightly higher eigenfrequencies (107%)
compared to their experimental counterparts. Model D
2
with standard
material values showed also good correlation with the experimental
mode shapes, although the predicted eigenfrequencies were higher
(111112%).
3.4. Reducing the detailed model of the large glulam truss
The detailed linear models consisting of solid elements were useful
Table 10
Comparing the eigenfrequencies of the paired experimental and numerical out-of-plane modes with MAC values >0.85 and with 50 DOFs.
f
EXP.OUT
(Hz)
f
D1
(Hz)
MAC
D1/EXP.OUT
f
D1
/f
EXP.OUT
(%)
f
D2
(Hz)
MAC
D2/EXP.OUT
f
D2
/f
EXP.OUT
(%)
2.3 2.7 0.93 120.3 2.8 0.98 124.0
2.7 2.8 0.97 102.9 3.0 0.98 109.8
5.4 6.2 0.97 114.1 6.5 0.98 119.2
6.4 6.8 0.98 107.2 7.3 0.99 113.8
12.0 12.5 0.99 103.9 13.2 0.99 110.0
13.5 13.8 0.98 102.7 14.7 0.98 109.0
17.8 18.8 0.86 105.6
18.9 19.5 0.95 103.4 20.6 0.97 108.8
21.3 22.1 0.85 103.5 22.9 0.94 107.5
22.2 23.3 0.85 104.8
24.3 25.0 0.92 102.5 25.7 0.90 105.7
26.4 27.5 0.86 104.2
31.8 34.3 0.88 107.8 35.1 0.87 110.4
35.3 40.5 0.85 114.6
Mean 0.94 106.7 0.93 110.9
Table 11
Comparing the eigenfrequencies of the paired experimental and numerical in-plane modes with MAC values >0.85 and with 38 DOFs.
f
EXP.IN
(Hz)
f
D1
(Hz)
MAC
D1/EXP.IN
f
D1
/f
EXP.IN
(%)
f
D2
(Hz)
MAC
D2/EXP.IN
f
D2
/f
EXP.IN
(%)
9.0 9.0 0.98 100.0 9.0 0.98 100.5
25.3 26.9 0.99 106.4 28.3 0.99 112.2
42.6 47.4 0.97 111.1 50.3 0.97 117.8
61.6 67.6 0.88 109.8 70.7 0.88 114.8
92.8 102.9 0.91 110.9 108.1 0.84 116.5
Mean 0.95 107.6 0.93 112.3
Fig. 10. Four reduction steps from the large and detailed model D
1
to the simple reduced model R
1
with some dimensions (in mm).
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
11
for predicting modal properties of the truss although those models had
several drawbacks. They were time consuming to develop and to
analyze, which make them unsuitable for the structural engineers daily
work. To improve the speed, several reduction techniques can be
applied, although the accuracy of the reduced models need to be
assessed and validated.
3.4.1. First reduced model with beams and small connection matrices
First, two methods were utilized to reduce the detailed large model
D
1
, namely there were (1) fewer and different types of nite elements
and (2) static condensation also known as Guyan-Irons reduction or
often just Guyan reduction. This reduced model is called R
1
. For the
glulam parts without any details and with coarser meshes, the solid el-
ements were replaced by beam elements including shear deformation
and rotatory inertia. The material model was changed to linear isotropic
with E according to Table 9, G =650 MPa and densities according to
Table 1. The Guyan method was used to reduce the stiffness and mass
matrices at the connection areas between the beam parts. Fourteen
connection models, each with between 0.45 and 2.26 millions DOFs,
were reduced to 2, 3 or 5 active nodes each. These nodes correspond to
the end nodes of the beam elements closest to the joints. Fig. 10 illus-
trates and describes in four steps the reduction rendering in model R
1
.
The Guyan reduction was performed using MSC Nastran and the
reduced stiffness and mass matrices, denoted K
red
and M
red
, were obtain
according to equations (3) and (4), with the remained active DOFs
denoted ‘aand the reduced dependent or interior DOFs ‘d.
K=Kaa Kad
Kda Kdd and T=I
K1
dd Kda (3)
Kred =TTKT and Mred =TTMT (4)
The active DOFs were constrained with rigid body elements (RBE2)
to the respective dependent DOFs that belong to the nodes of the solid
elements at the same cross section. The results from the modal analysis
of the reduced model R
1
and a comparison with the results from the
detailed model D
1
are presented in Table 12. The Guyan reduction is a
static condensation method, and it is exact for static analysis when in-
ternal DOFs are unloaded, but generally poor for modal analysis,
because the accuracy of predicting eigenfrequencies depends strongly
on the choice of the active and the dependent DOFs [43]. A Guyan
reduction results in a stiffer model, thereby a model with higher
eigenfrequencies. Meanwhile, the reduced model R
1
showed good re-
sults in comparison to the detailed model D
1
. Accuracy at predicting
modal properties with the reduced model was good for a wide range of
eigenfrequencies (2 to 96 Hz). The reduced model R
1
predicted the mode
shapes (MAC >0.9) and the eigenfrequencies very well, with 4% errors
for 18 eigenmodes compared to the detailed FE-model (Table 12).
Furthermore, model R
1
predicted the mode shapes (0.8 <MAC <0.9)
and the eigenfrequencies well, with 4% errors for 14 other eigenmodes
Table 12
Comparison of eigenfrequencies and mode shapes from the detailed model D
1
with the same stemming from the reduced model R
1
utilizing 780 DOFs for
modes having MAC values >0.9.
f
D1
(Hz)
f
R1
(Hz)
MAC
R1/D1
f
R1
/f
D1
(%)
6.8 7.0 0.91 102.2
9.0 9.3 1.00 103.2
13.8 14.1 0.91 101.7
19.5 19.7 0.91 100.7
25.0 25.0 0.96 100.4
26.9 28.0 0.93 104.2
28.3 28.5 0.96 100.5
35.7 35.6 0.95 99.6
39.3 39.3 0.95 100.0
44.4 44.5 0.95 100.3
61.4 61.2 0.96 99.7
64.6 64.9 0.96 100.4
67.6 68.0 0.94 100.6
71.8 71.9 0.98 100.2
79.1 79.0 0.94 99.8
81.0 81.4 0.92 100.4
87.8 87.8 0.91 100.0
96.0 96.1 0.99 100.1
Mean 0.95 100.8
Fig. 11. KT- and T-connection reduced to 3D-springs, rigid beams and new almost coincident nodes.
Table 13
Comparison of eigenfrequencies mode shapes from the detailed model D
1
with
the same stemming from the reduced model R
2
utilizing 780 DOFs for the elastic
modes having MAC values >0.8.
f
D1
(Hz)
f
R2
(Hz)
MAC
R2/D1
f
R2
/f
D1
(%)
2.7 2.6 0.90 97.0
2.8 2.8 0.91 98.6
6.8 6.5 0.93 94.7
9.0 9.2 1.00 102.1
12.5 12.0 0.87 96.4
13.8 13.5 0.92 98.0
19.5 18.2 0.91 93.2
26.9 29.0 0.94 107.9
35.7 35.1 0.87 98.1
39.3 36.8 0.88 93.5
47.4 44.9 0.82 94.7
71.8 73.5 0.84 102.4
116.3 108.1 0.86 92.9
Mean 0.90 97.7
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
12
compared to the detailed FE-model D
1
. From about 8,147,000 DOFs
(2,715,641 nodes) in the detailed FE-model D
1
, the total number of DOFs
was reduced to 1,566 (263 nodes) in the reduced model R
1
. Thereby, the
computational time for solving the same linear modal analysis with the
same hardware was reduced from about 1 h and 16 min down to about
40 s.
3.4.2. Second reduced model with springs and concentrated masses
The reduced model R
1
requires a solver that allows direct input of
stiffness and mass matrices, a feature seldom provided in structural FE-
software for civil engineers, and here, the reduced mass and stiffness
matrices of the connections were attempted to be exchanged to simple
connection models that can be used in readily available structural FE-
software. The simple models, together with the beam elements from
model R
1
, were developed and analyzed with MSC Nastran, and formed
a new reduced model called R
2
. For each of the 14 connections, new
almost coincident nodes, with up to 2 mm distance between them, were
created close to the intersection between the beams axes. The new
nodes were connected to respective end nodes of the beams (the same
nodes used in model R
1
) with a rigid beam element (MSC Nastran
RBAR). Then the new nodes were connected to each other via three-
dimensional (3D) springs (MSC Nastran CBUSH) with translational
and rotational stiffnesses. The springs six stiffnesses were obtained from
the diagonal terms of the stiffness matrices expressed locally using three
rotation matrices and reduced at the new almost coincident nodes. For a
T-connection, three new almost coincident nodes were created, the
reduced stiffness matrix had a size of 18 ×18 and it required three 3D-
springs to build model R
2
. For a KT-connection, ve new almost coin-
cident nodes were created, the reduced stiffness matrix had a size of 30
×30 and required ten 3D-springs (Fig. 11). The reduced mass matrix of
each connection was replaced by concentrated mass elements (CONM1)
placed at the new nodes. Each mass value was calculated as the sum of
the three diagonal translational mass terms stemming from the Guyan
reduction for a specic node multiplied by the ratio between the total
mass of the connection and the sum of the diagonal translational mass
terms for all reduced nodes in the connection.
The reduced model R
2
composed of rigid beams, 3D-springs and
concentrated masses connected to the beam elements was analyzed
quickly and can be implemented in commercially-available FE software
used by the building industry. Model R
2
was less accurate in predicting
modal properties than model R
1
, however, R
2
could very well (MAC >
0.9) predict the mode shapes and the eigenfrequencies with up to 8%
error for seven eigenmodes compared to the detailed FE-model D
1
.
Model R
2
could also quite well (MAC >0.8) predict the mode shapes and
the eigenfrequencies with up to 8% error for 13 elastic eigenmodes
compared to the detailed FE-model D
1
(Table 13).
3.4.3. Comparison to the Eurocode based model with springs
In a previous analyses of a beam model of the same large glulam
truss, 2D springs were used to model deformations at the connections
[24]. Values for two translational and one rotational stiffness were
evaluated using the slip modulus, K
ser
, for dowel shear joints according
to Eurocode 5 [33]. That model, called EC5 here, but A3 in the previous
paper, was used to calculate the in-plane eigenmodes. Table 14 presents
a comparison of these numerical results with the experimental results
[24, Table 3]. There was a good correlation between numerical and
experimental mode shapes and eigenfrequencies. To compare the
connection stiffness model to the detailed solid model, a static
condensation (Guyan method) was performed on the model EC5 at the
same active and dependent DOFs as for the reduction of model D
1
.
Reduced matrices of the same size were obtained for 13 connections
from both the detailed model and model EC5 (the connection C1 did not
use a 2D spring in the EC5 beam-model and was therefore not included
in the comparison). A screening of the ratios of the diagonal terms
corresponding to the DOFs in the plane of the truss (i.e., the translations
in Y and Z and the rotation around X), between models EC5 and R
1
,
showed values of around 0.67 with some ratios up to 1.5. The mean ratio
of the stiffness terms was almost the same for all 13 connections. The
ratios of mass diagonal terms for the same DOFs had average values
around 1.35. The frequencies predicted by model EC5 were just slightly
lower than the experimental ones and the connection stiffness values
had little impact on the frequencies.
4. Discussion
Several models of different sizes predicted the global dynamic
behavior of the large truss well. At the level of a glulam beam, the
calibration of the density and the longitudinal modulus of elasticity in a
solid model were more crucial than at the truss level. Indeed, the modal
results of the large, detailed models D
1
(with calibrated material prop-
erties) and D
2
(with standard material properties) were similar. Both
models were linear and made of many small solid elements. The
geometrical (gaps around the steel dowels and friction between the steel
and the timber parts) and material (hardening of the timber) non-
linearities that have been pointed out by many studies [20,3537]
even at low strain levels were not considered, still the detailed models
performed well to predict the dynamical responses on a large range of
eigenfrequencies.
The simplest reduction method, static condensation according to
Guyan and Irons, was implemented to reduce the connections and gave
pretty accurate modal results in less time. Several more accurate
methods exist, however, if the connections do not have a lot of internal
dynamics, the Guyan method which is simple to implement, and keeps
the physical DOFs, performs good enough. The Improved Reduced
System (IRS) and System Equivalent Reduction Expansion Process
(SEREP) methods are natural alternatives that could be investigated.
Here, the Guyan reduction method was used due to its simplicity and the
good accuracy demonstrated due to that the connections of the truss
were fairly rigid. By using mathematical reduction methods on detailed
models made of solid elements, less assumptions and estimations on the
contact and material properties are needed, and more general compar-
ative conclusions can be drawn between different models or between
numerical and experimental data.
In reduced truss models with beam elements, an exact evaluation of
the total mass is crucial [44]. The beam elements overlap in the joints,
thus the mass should be adjusted. According to the weighting of the 14
glulam members of the large truss investigated, the mean density per
member showed small differences compared to the mean standard
density for GL30c (430 kg/m
3
, Table 2), up to 6% (Table 1). Moreover,
addition of concentrated mass elements for the steel parts at the con-
nections was important.
A proper distribution of the stiffness is also important for predicting
the dynamic properties of the truss, and for a detailed model made of
solid elements, the stiffness values of the constitutive material model are
critical but not dependent on the geometry. According to the concise
sensitivity screening performed at the beam level, the longitudinal
modulus of elasticity was the main governing stiffness value among the
six parameters of the almost transversely isotropic material model.
However, a sensitivity study at the truss level can reveal the inuence of
Table 14
Comparison of the numerical and the experimental results for in-plane modes,
utilizing 24 DOFs.
f
EXP.IN
(Hz)
f
EC5
(Hz)
MAC
EC5/EXP.IN
f
EC5
/f
EXP.IN
(%)
9.0 9.0 1.0 100.0
25.3 24.2 1.0 95.7
42.7 42.1 0.99 98.6
61.6 60.0 0.79 97.4
92.8 84.7 0.92 91.3
Mean 0.94 96.6
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
13
the other ve stiffness parameters. The values of the longitudinal
modulus of elasticity of the glulam members showed large differences
compared to the mean standard value for GL30c (12.3 GPa, Table 2), up
to 17% (Table 9), which is the highest difference accepted according to
[38].
The second reduced model R
2
with concentrated masses, rigid beams
and 3D-springs with six stiffness values each was analyzed quickly, and
it was adapted to structural engineersFE-tools, but it had poorer modal
prediction than the reduced model R
1
. It was not capable of perfectly
mimicking the associated portions of the reduced matrices; making a
perfect mimic would have required more independent stiffness and mass
parameters. However, the intention here was merely to show that such a
model strategy is possible and can be accomplished with somewhat
more advanced models, if FE-software already in use at a company is to
be used. Furthermore, the level of accuracy should be evaluated in
relation to the purpose of the model. According to the methods used in
building codes to mitigate the wind-induced vibration, only the rst
eigenmode in the direction of the sway is noteworthy. Thus, the model
R
2
might be adequate for this purpose.
The Guyan method was used to assess the simple beam model EC5
that was developed in a previous study using common engineering
praxis, calibrated material properties and the slip modulus of the
Eurocode 5 to evaluate the translational and rotational stiffness at the
connections made of 2D-springs. The reduced mass and stiffness
matrices were then compared to the reduced matrices obtain from the
detailed model D
1
and indicated that the EC5 model is underestimating
the stiffness of the connections, in the plane, with a factor of about two
third. This conrms results from static tests on dowel connections pre-
sented in [20,23,35]. Meanwhile, the model EC5 revealed very good
accuracy to predict the eigenfrequencies of the in-plane modes although
they were lower than the measured ones.
5. Conclusion
The longitudinal stiffness moduli of glulam beams evaluated with
modal testing and measured densities were used to develop a detailed
FE-model of a large glulam truss with steel plates and dowels connec-
tions. The linear model with calibrated mass and stiffness parameters
could accurately predict the modal properties of the truss for a wide
range of frequencies both in the plane and out of the plane of the truss.
The use of standard values for stiffness and densities of glulam induced
only small difference. To reduce the calculation time, a reduced model
was obtained by changing some solid elements to beam elements and by
static condensation, with the Guyan reduction method, of the stiffness
and mass matrices at the connections. The CPU time for modal analysis
were strongly reduced while very good predictive accuracy was
retained. An engineer-friendly and reduced model with springs con-
necting the beam elements and concentrated masses at the joints was
developed, and the accuracy was lower but sufcient. By implementing
a parameterized detailed part, e.g., a connection, in the accurate
reduced model, extensive sensitivity analysis could be performed. Sub-
sequently, the impacts of different parameters (such as the nonlinear
embedment of dowels) could be investigated further in order to better
understand and predict the dynamical response of large glulam trusses
with slotted in steel plates and dowels connections.
CRediT authorship contribution statement
Pierre Landel: Writing original draft, Conceptualization, Meth-
odology, Software, Validation, Formal analysis, Investigation, Re-
sources, Data curation, Writing review & editing, Visualization,
Project administration, Funding acquisition. Andreas Linderholt: Su-
pervision, Conceptualization, Methodology, Software, Validation,
Formal analysis, Investigation, Resources, Data curation, Writing re-
view & editing, Visualization, Project administration, Funding
acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing nancial
interests or personal relationships that could have appeared to inuence
the work reported in this paper.
Acknowledgments
This research received funding from the ForestValue Research Pro-
gram, which is a transnational research, development and innovation
program jointly funded by national funding organizations within the
framework of the ERA-NET Cofund ‘ForestValue Innovating forest-
based bioeconomy. The Swedish part of the Project Dyna-TTB is sup-
ported under grant No 2018-04976 by Swedish Governmental Agency
for Innovation Systems (Vinnova), The Swedish Research Council for
Environment, Agricultural Sciences and Spatial Planning (FORMAS) and
Swedish Energy Agency (SWEA). Moelven T¨
oreboda AB is deeply
thanked for the opportunity to do this research at their factory and for
their kind assistance in handling the glulam truss.
References
[1] Ohlsson S. Floor vibrations and human discomfort. Division of Steel and Timber
Structures: Chalmers University of Technology; 1982.
[2] Smith I, Chui YH. Design of lightweight wooden oors to avoid human discomfort.
Can J Civ Eng 1988 Apr 1;15(2):25462.
[3] Landel P, Linderholt A and Johansson M. Comparative study of wind-induced
accelerations in tall timber buildings according to four methods. World Conference
on Timber Engineering, August 9-12, 2021, Santiago, Chile.
[4] Bjertnes MA and Malo KA. Wind.-induced motions of Treet” – a 14-storey timber
residential building in Norway. World Conference on Timber Engineering, Quebec
Canada 2014.
[5] Hu L, Omeranovic A, Gagnon S and Mohammad SM Wind-induced vibrations of
tall wood buildings is it an issue?In: World Conference on Timber Engineering,
Quebec City 2014.
[6] Johansson M, Linderholt A, Bolmsvik Å, Jarner¨
o K, Olsson J and Reynolds T.
Building higher with light-weight timber structures the effect of wind induced
vibrations, Proceedings of the Internoise 2015, 10-13 August, San Francisco, USA.
[7] Malo KA, Abrahamsen RB, Bjertnaes MA. Some structural design issues of the 14-
storey timber framed building Treetin Norway. Eur J Wood Wood Prod 2016;74
(3):40724. https://doi.org/10.1007/s00107-016-1022-5.
[8] Abrahamsen R. Mjøstårnet-Construction of an 81 m tall timber building.
Internationales Holzbau-Forum IHF 2017:112.
[9] Lazzarini E, Frison G, Trutalli D, Marchi L, Scotta R. Comfort assessment of high-
rise timber buildings exposed to wind-induced vibrations. The Structural Design of
Tall and Special Buildings:e1882 2021;30(12). https://doi.org/10.1002/tal.1882.
[10] Johansson M, Linderholt A, Jarner¨
o K and Landel P. Tall timber buildings - a
preliminary study of wind-induced vibrations of a 22-storey building. In World
Conference on Timber Engineering, 2016, Wien, Austria.
[11] Edsk¨
ar I, Lidel¨
ow H. Wind-induced vibrations in timber buildingsparameter
study of cross-laminated timber residential structures. Structural Engineering
International 2017;27(2):20516. https://doi.org/10.2749/
101686617X14881932435619.
[12] Edsk¨
ar I, Lidel¨
ow H. Dynamic properties of cross-laminated timber and timber
truss building systems. Eng Struct 2019;186:52535. https://doi.org/10.1016/j.
engstruct.2019.01.136.
[13] Bezabeh MA, Bitsuamlak GT, Popovski M, Tesfamariam S. Dynamic response of tall
mass-timber buildings to wind excitation. J Struct Eng 2020 Oct 1;146(10):
04020199. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002746.
[14] Brancheriau L, Baill`
eres H. Natural vibration analysis of clear wooden beams: a
theoretical review. Wood Sci Technol 2002 Aug;36(4):34765. https://doi.org/
10.1007/s00226-002-0143-7.
[15] Steiger R, Gülzow A, Czaderski C, Howald MT, Niemz P. Comparison of bending
stiffness of cross-laminated solid timber derived by modal analysis of full panels
and by bending tests of strip-shaped specimens. Eur J Wood Wood Prod 2012 Jan;
70(1):14153. https://doi.org/10.1007/s00107-011-0521-7.
[16] Labonnote N, Rønnquist A, Malo KA. Experimental evaluations of material
damping in timber beams of structural dimensions. Wood Sci Technol 2013 Sep 1;
47(5):103350. https://doi.org/10.1007/s00226-013-0556-5.
[17] Yu LL, Wang J and Xia D. Vibration Method for Elastic Modulus of Glued
Laminated Beams. In 3rd International Conference on Material, Mechanical and
Manufacturing Engineering (IC3ME 2015) 2015 Aug (pp. 1603-1606). Atlantis
Press.
[18] Bui TA, Oudjene M, Lardeur P, Khelifa M, Rogaume Y. Towards experimental and
numerical assessment of the vibrational serviceability comfort of adhesive free
laminated timber beams and CLT panels assembled using compressed wood
dowels. Eng Struct 2020 Aug;1(216):110586. https://doi.org/10.1016/j.
engstruct.2020.110586.
P. Landel and A. Linderholt
Engineering Structures 260 (2022) 114208
14
[19] Reynolds T, Harris R, Chang WS. Stiffness of dowel-type timber connections under
pre-yield oscillating loads. Eng Struct 2014 Apr;15(65):219. https://doi.org/
10.1016/j.engstruct.2014.01.024.
[20] Reynolds T, Miranda W, Trabucco D, Toumpanaki E, Foster R, Stiffness RM, et al.
World Conference on Timber Engineering, August 2023, 2018. Seoul: Republic of
Korea; 2018.
[21] Heiduschke A, Kasal B, Haller P. Analysis of wood-composite laminated frames
under dynamic loadsanalytical models and model validation. Part I: connection
model. Prog Struct Mat Eng 2006 Jul;8(3):10310. https://doi.org/10.1002/
pse.217.
[22] Heiduschke A, Kasal B, Haller P. Analysis of wood-composite laminated frames
under dynamic loadsanalytical models and model validation. Part II: frame
model. Prog Struct Mat Eng 2006 Jul;8(3):1119. https://doi.org/10.1002/
pse.218.
[23] Solarino F, Giresini L, Chang WS, Huang H. Experimental tests on a dowel-type
timber connection and validation of numerical models. Buildings 2017 Dec;7(4):
116. https://doi.org/10.3390/buildings7040116.
[24] Landel P, Linderholt A and Johansson M. Dynamical properties of a large glulam
truss for a tall timber building. In 2018 World Conference on Timber Engineering,
August 20-23, 2018, Seoul, Republic of Korea.
[25] Reynolds T, Harris R, Chang WS, Bregulla J, Bawcombe J. Ambient vibration tests
of a cross-laminated timber building. In Proceedings of the Institution of Civil
Engineers - Construction Materials 2015;168(3):12131. https://doi.org/10.1680/
coma.14.00047.
[26] Feldmann A, Huang H, Chang W, Harris R, Dietsch P, Gr¨
afe M and Hein C. Dynamic
properties of tall timber structures under wind-induced vibration. In 2016 World
Conference on Timber Engineering. August 2016, Wien, Austria.
[27] Auclair SC, Hu L, Gagnon S, and Mohammad SM. Effect of type of lateral load
resisting system on the natural frequencies of mid- to high-rise wood buildings. In
2018 World Conference on Timber Engineering, August 20-23, 2018, Seoul,
Republic of Korea.
[28] Mugabo I, Barbosa AR, Riggio M. Dynamic characterization and vibration analysis
of a four-story mass timber building. Frontiers Built Environ 2019 Jul;4(5):86.
https://doi.org/10.3389/fbuil.2019.00086.
[29] Steiger R, Feltrin G, Weber F, Nerbano S, Motavalli M. Experimental modal
analysis of a multi-storey light-frame timber building. Bull Earthq Eng 2017 Aug;15
(8):326591. https://doi.org/10.1007/s10518-015-9828-9.
[30] Abrahamsen R, et al. Dynamic response of tall timber buildings under service load
-the DYNATTB research program. In EURODYN 2020, XI International Conference
on Structural Dynamics. November 23-26 2020, Athens, Greece.
[31] Kurent B, Brank B, Ao WK. Model updating of seven-storey cross-laminated timber
building designed on frequency-response-functions-based modal testing. Struct
Infrastruct Eng 2021 May;19:19. https://doi.org/10.1080/
15732479.2021.1931893.
[32] Manthey M, et al. Effect of non-structural components on natural frequency and
damping of tall timber building under wind loading. In 2021 World Conference on
Timber Engineering, August 9-12, 2021, Santiago, Chile.
[33] CEN. Eurocode 5: Design of timber structures part 1-1: General common rules
and rules for buildings. European Standard EN 1995-1-1, Comit´
e Europ´
een de
Normalisation, 2004.
[34] Vilguts A, Stamatopoulos H, Malo KA. Parametric analyses and feasibility study of
moment-resisting timber frames under service load. Eng Struct 2021;228:111583.
https://doi.org/10.1016/j.engstruct.2020.111583.
[35] Sandhaas C. Mechanical behaviour of timber joints with slotted-in steel plates.
Doctoral thesis. TU Delft, Netherlands; 2012.
[36] Dorn M. Investigations on the serviceability limit state of dowel-type timber
connections. Doctoral thesis. TU Wien, Austria; 2012.
[37] Reynolds T. Dynamic behaviour of dowel type connections under in service
vibration. UK: University of Bath; 2013. Doctoral thesis.
[38] European committee for standardization: EN 14080:2013 Timber structures-
glued laminated timber and glued solid timber Requirements. CEN, Brussels,
Belgium; 2013.
[39] European committee for standardization: EN 338:2016 Structural timber
Strength classes. CEN, Brussels, Belgium; 2013.
[40] Carne TG, Grifth DT, Casias ME. Support conditions for experimental modal
analysis. Sound and Vibration 2007 Jun 1;41(6):106.
[41] International Organization for Standardization: ISO 10137:2007 Bases for design
of structures Serviceability of buildings and walkways against vibration. Geneva,
Switzerland; 2007.
[42] Blaß HJ and Bejtka I. Numerische Berechnung der Tragf¨
ahigkeit und der Steigkeit
von querzugverst¨
arkten Verbindungen mit stiftf¨
ormigen Verbindungsmitteln.
Karlsruher Berichte zum Ingenieurholzbau Band 10, Universit¨
atsverlag Karlsruhe;
2008.
[43] Craig Jr RR, Kurdila AJ. Fundamentals of structural dynamics. John Wiley & Sons;
2006.
[44] Landel P, Linderholt A and Johansson M. Test-analyses comparisons of a stabilizing
glulam truss for a tall building. CompWood. 2019, 30:11, V¨
axj¨
o, Sweden.
P. Landel and A. Linderholt
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