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Homogenized material properties for cross‐laminated timber
Article in PAMM · December 2018
DOI: 10.1002/pamm.201800037
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2. Received: 14 May 2018 Accepted: 26 August 2018
DOI: 10.1002/pamm.201800037
Homogenized material properties for cross-laminated timber
Thomas Furtmüller1,∗
, Christoph Adam1
, and Benjamin Giger1
1
University of Innsbruck, Unit of Applied Mechanics, Technikerstraße 13, 6020 Innsbruck, Austria
In this contribution the coefficients of the elastic stiffness tensor of a generalized shell section for cross-laminated timber
(CLT) slabs are derived by application of a numerical homogenization scheme based on a repeating unit cell (RUC). In an
example it is shown that a finite shell element model with those homogenized shell stiffness parameters predicts closely
the modal properties of a CLT slab, thus, verifying the accuracy of the applied homogenization procedure for this type of
structural members.
c 2018 The Authors. PAMM published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim.
1 Introduction
Cross-laminated timber (CLT) is a planar structural element, consisting of several layers of timber boards glued on their wide
faces but not necessarily on their narrow faces. The layers are stacked perpendicular to each other, with the outer layers
having the same orientation. Through this design, CLT is a member with enhanced stiffness and strength, and favorable
shrinking/swelling behavior compared to regular timber, whose mechanical properties in grain direction are significantly
different from across the grain. CLT can be used in panel-like structures such as walls and slabs, making structural timber
an appealing alternative to steel or concrete structures. Nowadays, in engineering practice, the finite element method is used
to predict the response and load-bearing capacity of structures, based on a discretization by shell elements in case of planar
structural members. For layered structures, shell elements can only be used if homogenized cross-sectional stiffness properties
are available. In this contribution, the homogenized parameters of a shell stiffness tensor for CLT composed of five layers of
spruce wood with equal layer thickness of 20 mm are derived, see also [1].
Fig. 1: Experimental (black) and numerical (red) load
displacement curves of three-point bending tests on CLT [1].
Fig. 2: Repeating unit cell for numerical
homogenization of CLT [1].
2 Material parameters of the timber boards
Before the homogenized behavior of CLT is addressed, the material parameters of the individual timber layers need to be
defined and identified. To this end, three-point bending tests on CLT specimens with length of 110 cm and width of 40 cm
are conducted. The three considered CLT setups differ in the orientation of the layers. Subsequently, numerical optimization
on a finite element model of these specimens (discretized by solid elements) delivers the parameters of the constitutive laws
assumed to capture the behavior of the timber boards. As reference serve the experimental load-displacement relations. In
the numerical model the spruce boards are assumed to be linearly elastic orthotropic-perfectly plastic. Elastic orthotropy is
characterized by nine independent material parameters. The plastic behavior is described by Hill’s yield criterion [2] captured
by six constitutive parameters. In Hill’s yield criterion different yield strengths can be assigned in three perpendicular material
orientations, as observed in timber. However, the strength properties in compression and tension are the same, which is in
contradiction to the actual behavior of wood. An additional parameter to be defined is the friction coefficient between the
narrow faces of the boards. The, in total, 16 parameters determined in the optimization procedure are summarized in [1].
∗ Corresponding author: email thomas.furtmueller@uibk.ac.at, phone +43 512 507 61617
This is an open access article under the terms of the Creative Commons Attribution License 4.0, which permits use,
distribution and reproduction in any medium, provided the original work is properly cited.
PAMM · Proc. Appl. Math. Mech. 2018;18:e201800037. www.gamm-proceedings.com c 2018 The Authors. PAMM published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim. 1 of 2
https://doi.org/10.1002/pamm.201800037

3. 2 of 2 Section 4: Structural mechanics
As an outcome, Fig. 1 shows the experimental load-displacement curves of the CLT specimens and the numerically derived
counterparts based on the identified material parameters of the spruce boards and on the friction coefficient, which are in
excellent agreement. For details on the derivation of the constitutive parameters it is referred to [1].
3 Homogenization scheme
The cross-sectional stiffness parameters of a generalized shell stiffness tensor are derived by conducting numerical homog-
enization on a repeating unit cell (RUC) of the considered CLT setups, depicted in Fig. 2. To this end, a scheme developed
in [3] and refined in [4] for numerical homogenization of masonry walls is applied. That is, to the RUC successively unit
macroscopic strains and curvatures are imposed, yielding the stiffness coefficient of the stiffness tensor. Note that, although
linear elastic behavior of the timber layers is assumed, the overall behavior of the RUC is nonlinear due to the presence of
Coulomb friction between the narrow faces of the boards. Thus, in this sense the outcomes are linearized homogenized elastic
properties. The resulting elastic homogenized shell parameters as well as a discussion on the effect of friction and plasticity
in the RUC are found in [1].
4 Application and verification
In an application example, the modal parameters of a point-supported CLT slab are derived numerically. The overall dimen-
sions of the L-shaped slab are 10.7 × 10.7 m, with a distance between the supports of 5.1 m. In the first approach, the slab is
discretized by finite shell elements, and the derived homogenized CLT shell parameters are assigned to the model. Since this
finite shell element model exhibits 460.000 degrees of freedom only, modal analysis can be conducted on a personal computer.
As reference solution serve the outcomes of a more elaborate finite solid element model, employing hexahedral solid elements
with 12.2 million degrees of freedom. Modal analysis of this model requires 246 GB of RAM, and thus, can only be conducted
on a high-performance computer cluster. Fig. 3 shows the second mode shape of the computationally expensive solid element
model, and Fig. 4 its approximation by the shell element model. The difference of the corresponding natural frequency of
the latter model, f
(shell)
2 = 7.44 Hz, and the reference solution, f
(solid)
2 = 7.36 Hz, is only 1%. The correlation between
the mode shapes of the solid and the shell element model is further quantified by the modal assurance criterion (MAC) [5].
According to Fig. 5, the MAC value of the first ten modes is close to one. That is, for the considered bending-dominated
structure the simplified shell finite element model with assigned homogenized stiffness parameters predicts closely the actual
modal slab properties, thus, verifying the applicability and accuracy of the employed homogenization scheme for CLT slabs.
Fig. 3: Second mode shape of the solid
element model, fsolid
2 = 7.36 Hz [1].
Fig. 4: Second mode shape of the shell
element model, f
(shell)
2 = 7.44 Hz [1].
Fig. 5: Modal assurance criterion matrix for
solid element and shell element model [1].
Acknowledgements The computational results presented have been achieved in part using the HPC infrastructure LEO of the University
of Innsbruck. This work has been supported as part of the research project Spider Connector [project number 223803] funded by the
Austrian Research Promotion Agency (FFG).
References
[1] T. Furtmüller, B. Giger, and C. Adam, Eng Struct 163, 77 (2018).
[2] R. Hill, Proc Roy Soc A 193, 281 (1948).
[3] M. Mistler, A. Anthoine, and C. Butenweg, Comput Struct 85, 1321 (2007).
[4] T. Furtmüller and C. Adam, Acta Mech 221, 65 (2011).
[5] R. J. Allemang and D. L. Brown, Proceedings 1st International modal analysis conference, Orlando, FL (1982), pp. 110–116
c 2018 The Authors. PAMM published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim. www.gamm-proceedings.com
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