Multi-scale model updating for the mechanical properties of cross-laminated timber

Computers and Structures 177 (2016) 83–90

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Multi-scale model updating for the mechanical properties of crosslaminated timber E.I. Saavedra Flores a,⇑, R.M. Ajaj b, I. Dayyani c, Y. Chandra d, R. Das e a

Departamento de Ingeniería en Obras Civiles, Universidad de Santiago de Chile, Av. Ecuador 3659, Estación Central, Santiago, Chile Aeronautics and Astronautics, University of Southampton, Southampton SO171BJ, UK c Centre for Structures, Assembly and Intelligent Automation, School of Aerospace, Transport and Manufacturing, Cranfield University, Cranfield MK43 0AL, UK d College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK e Sir Lawrence Wackett Aerospace Research Centre, School of Engineering, RMIT University, GPO Box 2476, Melbourne, VIC 3001, Australia b

a r t i c l e

i n f o

Article history: Received 28 July 2015 Accepted 31 August 2016

Keywords: Cross-laminated timber Multi-scale modelling Finite elements Genetic Algorithm

a b s t r a c t In this paper we propose a homogenisation-based four-scale model for the mechanical properties of cross-laminated timber. The spatial scales considered in this study are the wood cell-wall, the wood fibres, growth rings and the structural scale. The computational homogenisation scheme is solved sequently from the lowest to the highest level in order to determine the effective mechanical properties of each material scale. As we are interested in improving the predictions of our computational simulations, we propose an optimisation strategy to calibrate the micro-mechanical parameters. Our numerical predictions are compared with experimental results and are validated successfully. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Over the last two decades or so, cross-laminated timber (CLT) has been gaining popularity in residential applications, mainly in Europe and North America. CLT is a relatively new building system based on structural panels made of several layers of boards stacked crosswise and glued together on their faces (Fig. 1). As CLT panels are light-weight structural elements with high stiffness and strength to bending, compression and shear, they are an economically competitive building system when compared to traditional options and therefore, are a suitable candidate for some applications which currently use concrete, masonry and steel [2]. CLT has multiple advantages including its favourable seismic performance, its ability to self-protect against fire, its lessened environmental impact and its renewable material source [3]. In spite of the advantages of building with CLT, and the considerable growth that the total production is experiencing in the world market [4], the benefits of using CLT and in general timber, in the construction industry are still far from maximised. This is mainly due to the fact that dimensioning practices and many existing structural design rules are still based on an empirical background [5]. Different methods have been adopted for the determination of the basic mechanical properties of CLT. However,

⇑ Corresponding author. E-mail address: [email protected] (E.I. Saavedra Flores). http://dx.doi.org/10.1016/j.compstruc.2016.08.009 0045-7949/Ó 2016 Elsevier Ltd. All rights reserved.

to date no method has been universally accepted by CLT manufacturers and designers [6]. The reason for the slow progress in the development of timber design codes, and in particular, in the difficulties to fully understand the mechanics of timber materials, lies mainly in the highly complex and intricate nature of wood microstructure [7]. At very small scales, wood shows a complicated hierarchical nature distributed across multiple spatial scales, from submicrometer dimensions to macroscopic scales. This important feature has been a subject of intensive research over the last few years by means of multi-scale homogenisation techniques. Initial investigations were carried out by Holmberg et al. [8] on the mechanical behaviour of wood from a micro up to a macro level. They obtained numerically stiffness and shrinkage properties and compared them with experimental data. Hofstetter et al. [9] suggested five elementary phases for the mechanical characterisation of wood. These were hemicellulose, lignin, cellulose, with its crystalline and amorphous portions, and water. Qing and Mishnaevsky [10] studied the effect of wood density, microfibril angle (MFA) and cell shape on the longitudinal tensile strength of softwood. Rafsanjani et al. [11,12] investigated experimentally and numerically the hygroscopic swelling and shrinkage properties of softwood. Saavedra Flores and Friswell [13] investigated the deformation and failure mechanisms of wood at the ultrastructural scale. They also studied the development of a new material inspired by the mechanics and structure of wood cell-walls [14]. In the context of multi-scale modelling of CLT

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Fig. 1. Schematic representation of a typical CLT panel [1].

structures, Saavedra Flores et al. [15,7] investigated the structural behaviour of CLT by linking three different scales. We note, however, that despite the increasing interest in this subject, the complete understanding of the mechanical properties of wood, and in particular cross-laminated timber, is still an issue which remains open at present. In this new paper, we continue with the line of development started in the above references [15,7] by introducing the following new features: 1. The explicit modelling of growth rings as a new material scale. 2. Multi-scale model updating by means of a Genetic Algorithm (GA). 3. Investigation of the influence of wood density on the mechanical properties of CLT. 4. New experimental results. This paper is organised as follows. Section 2 describes briefly the mathematical foundations of the multi-scale constitutive theory. Section 3 presents the strategy adopted for the multi-scale modelling of timber structures. The experimental works are described in detail in Section 4. Section 5 introduces the use of the GA technique to improve our numerical predictions. The validation of our (updated) model along with some numerical predictions are presented in Section 6. Finally, Section 7 summarises our main conclusions. 2. Multi-scale constitutive theory Multi-scale models enable specifying the relationships between physical variables observed at different length scales. These are of particular importance in the study of heterogeneous materials with hierarchical microstructures in which the macroscopic response of the material can be predicted from the information coming from the microscopic (or lower) level. In the present type of homogenisation-based multi-scale constitutive theory, each material scale is associated with a microstructure whose most statistically relevant features are incorporated within a representative volume element (RVE). This RVE is assumed to have a (microscopic) characteristic length much smaller than the macro-continuum, and at the same time, a size large enough to capture the microscopic heterogeneities in an averaged sense. In this theory it is also assumed that the macroscopic or homogenised strain tensor component eij at any arbitrary point of the macroscopic continuum is the volume average of the microscopic

strain tensor component el ij over the domain Xl of the RVE. Similarly, the macroscopic or homogenised stress tensor component rij is assumed to be the volume average of the microscopic stress tensor component rl ij over Xl . In addition, we can introduce a convenient decomposition in the total displacements field over the RVE domain as a sum of a lin~ li ear displacement component and a displacement fluctuation u which represents local variations about the linear displacement component and does not contribute to the macroscopic scale strain. By taking into account the Hill-Mandel Principle of Macrohomogeneity [16,17], which establishes that the macroscopic stress power must equal the volume average of the microscopic stress power over Xl , the virtual work equation for the RVE can be reduced to

Z

Xl

Z

rl ij deij dV ¼

Xl

rl ij ð@ gi =@yj þ @ gj =@yi Þ=2 dV ¼ 0;

ð1Þ

with deij representing the Cartesian components of the kinematically admissible virtual strains field, gi a component of the virtual displacements vector, both at the RVE level, and yi a local (RVE) coordinate. We note in Eq. (1) that, the Hill-Mandel Principle requires the RVE body force and external surface traction fields to produce no virtual work [18]. As it stands, Eq. (1) leads to an ill-posed microscopic equilibrium problem. Therefore, in order to make problem (1) wellposed, we introduce in the present multi-scale constitutive framework a set of kinematical constraints on the displacement fluctuations (belonging to the vector space of virtual displacements) to be imposed in the RVE. The choice of different kinematical constraints defines different classes of multi-scale constitutive models. Probably the most popular classes are the Taylor model, or rule of mixtures, and the Periodic boundary displacement fluctuations model. The Taylor model is obtained by setting to zero all the displacement fluctuations in the RVE domain. This choice implies that the corresponding total microscopic displacement field varies linearly in Xl and that the microscopic strain field is homogeneous. One important drawback of the Taylor kinematical constraint is the fact that it does not consider the mechanical interaction among the different solid phases or between the solid phases and micro-voids. For this particular class of model, the homogenised constitutive tangent operator is calculated by taylor Dhom ¼ ijkl ¼ Dijkl

1 Vl

Z

Xl

Dl ijkl dV;

ð2Þ

E.I. Saavedra Flores et al. / Computers and Structures 177 (2016) 83–90

where Dl ijkl are the Cartesian components of the microscopic constitutive tangent operator. That is, for the Taylor model, the homogenised tangent tensor is the volume average of the microscopic constitutive tangent tensor. The Periodic boundary displacement fluctuations model is typically chosen for the description of periodic media and will be adopted for all our computational simulations because of the periodic nature of wood’s microstructure (as discussed later in Section 3). In this class of model, the periodic repetition of the RVE generates the entire heterogeneous macro-continuum. The kinematical assumption for this class of constitutive model consists of prescribing identical displacement fluctuation vectors for each pair of opposite points on the boundary @ Xl of the RVE domain [19]. In a general case, the homogenised constitutive tangent operator can be expressed as a sum of the Taylor constitutive tangent e ijkl . This relaoperator and a tangential fluctuations contribution, D tion is expressed as taylor e ijkl : Dhom þD ijkl ¼ Dijkl

ð3Þ

The derivation of the corresponding tangential fluctuations contribution is extensive and falls outside the scope of this paper. However, for those readers interested in the details of this calculation, we refer to [18,20,21]. For the Periodic boundary displacement fluctuations model, the above general expression (3) holds along with the corresponding periodic kinematical constraint. Finally, the computational homogenisation procedure described in this section is implemented in the commercial software ANSYS 15.0 [22] in order to perform all of our finite element-based multi-scale simulations. 3. Multi-scale modelling strategy for timber In this section, we describe the multi-scale finite element modelling of timber. The type of wood chosen for this study is radiata pine, which has several applications in building and engineering structures. As commented in the previous section, the class of multi-scale model adopted here corresponds to the Periodic boundary displacement fluctuations model. The procedure described in the following consists of modelling the mechanical response of the CLT structure by means of four fundamental spatial scales. These are the wood cell-wall at the order of few nanometers, the wood fibres with cross section dimensions of tens of micrometers, growth rings described by some few millimeters and the structural scale, at the order of meters. We note that the first two material scales have already been described in [15] and therefore, we skip the details about their modelling. At the wood cell-wall’s scale, wood contains three fundamental phases: cellulose, hemicellulose and lignin. These three fundamental constituents form the wood cell-wall composite material whose basic unit building block is called microfibril. This composite comprises reinforcing micro-fibres which are oriented mainly in a single direction (in almost the whole cell-wall’s volume) and are embedded periodically in a softer matrix. The reinforcing microfibre is made up of periodic alternations of crystalline and amorphous cellulose fractions. The length of each crystalline cellulose portion is hereafter referred to as Lcc . The degree of cellulose crystallinity, f cc , is defined as the volume fraction of the crystalline portion of cellulose with respect to the total volume of cellulose. The volume of (crystalline and amorphous) cellulose with respect to the total volume of cell-wall is termed f c . The matrix of the cellwall composite is made up of hemicellulose and lignin polymers. Hemicellulose is built up of sugar units and has little strength, with mechanical properties highly sensitive to moisture changes. The

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volume fraction of hemicellulose with respect to the total volume of cell-wall is referred to as f h . Lignin is an amorphous and hydrophobic polymer and its main purpose is to cement the individual wood fibres together and to provide inter-fibre shear strength. The specific orientation of the microfibrils with respect to the wood fibre’s axis is called the microfibril angle, MFA. A typical finite element mesh of the RVE associated with this scale is shown in Fig. 2(a). Further details can be found in [15]. At the next scale, the material can be represented by a periodic arrangement of long slender tubular fibres, oriented nearly parallel to the axis of the stem. The cross-sections of each wood fibre is (normally) hexagonal, and can be defined by means of four geometric parameters. These are the tangential and radial dimensions of the hexagonal cross-section, denoted here as T and R, respectively, the thickness of the (tubular) cross-section, t, and the angle h (whose value can be, for instance, 0° for a rectangular crosssection, or 30° for a regular hexagonal shape). In softwoods, wood fibres can be divided into early-wood and late-wood. The earlywood fibres are characterised by large diameters and thin cellwalls, whereas late-wood fibres are composed of narrow diameters with much thicker cell-walls. Fig. 2(b) and (c) shows typical finite element meshes for the RVEs associated with late-wood and earlywood, respectively. The former has 3141 nodes and 1840 elements, and the latter 1626 nodes and 672 elements. In all of our simulations, we select the eight-noded SOLID185 element type in ANSYS. Uniform reduced integration technique with hourglass stiffness for controlling hourglass modes has been chosen in order to prevent shear locking under bending loading conditions. In both figures, the green colour indicates the (P+M)-layer, whereas the red colour shows the S2 -layer whose mechanical properties are obtained by means of the computational homogenisation of the microfibril RVE shown in Fig. 2(a). In this homogenisation procedure, the MFA has been taken into account. Details on the modelling of the (P+M)-layer can be found in [15]. For further information about the morphology and composition of wood at the nano- and microscopic scale level (and in particular, on the wood cell-wall layers), we refer, for instance, to [23,24]. The following scale is represented by the growth rings, typically found in the cross-section cut through the trunk of a tree. Each growth ring exhibits two colour regions associated with earlywood (light colour) and late-wood (dark colour) fibres. Within a growth ring, the volume fraction of early-wood fibres with respect to the total volume of growth ring is denoted as f ew . A typical finite element mesh of the RVE chosen to describe the mechanical response of the growth ring (called here the growth ring RVE) is shown in Fig. 2(d). It consists of 288 nodes and 165 hexahedral elements. The turquoise colour represents the portion of material calculated by the computational homogenisation of the early-wood RVE shown in Fig. 2(c), whereas the light brown colour shows the material obtained by the homogenisation of the latewood RVE shown in Fig. 2(b). The periodic repetition of the growth rings forms the base material for the macroscopic or structural scale. At this level, we perform several experimental tests on wooden specimens which will be described in detail in the next Section 4.

4. Experimental works The type of wood chosen for this study was Chilean radiata pine and the adhesive used to manufacture the specimens was EPI (Emulsion Polymer Isocyanate) system PREFERE 6151/6651 [25]. Budget constraints allowed us to perform only a limited number of experiments. In the following we described these tests.

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Fig. 2. Finite element meshes of the RVEs and structures analysed in this paper along with their corresponding length scales. (a) RVE associated with the microfibril scale (for the sake of clarity, only one half of the RVE is shown here); (b) RVE associated with the modelling of late-wood fibres; (c) RVE associated with early-wood fibres; (d) growth ring RVE; (e) 4-cm-thick layer subject to four-point bending; (f) 12-cm-thick CLT panel (consisting of three 4-cm-thick layers) subject to three-point bending.

1. Preliminary experimental tests on individual timber pieces. We conducted experimental tests to measure density and moisture content in 103 individual timber pieces employed for the manufacture of the specimens. The experimental measurements were carried out according to the Chilean standard NCh 176 Of. 2003 [26]. The moisture content was measured in several points in each individual piece and its average value was 11%. This allowed us to assume dry conditions in all of our multi-scale simulations. The average density for all the pieces was 506 kg=m3 . 2. Wooden layers subject to four-point bending along the longitudinal direction. In order to determine the longitudinal Young’s modulus we tested six 4-cm-thick layers subject to four-point bending. Each layer was made up of individual wooden pieces glued together on their edges and oriented in the long direction of the specimen, as shown in Fig. 3(a). The total length and width of the layer were 2.4 m and 1.2 m, respectively. The central span length between supports was 2.25 m and the distance between vertical loads was 1.125 m. We measured experimentally the longitudinal Young’s modulus in each specimen and we obtained an average value of 12,712 MPa. We note that this

value along with the average experimental density of 506 kg=m3 will be of great importance in the multi-scale model updating process described later in Section 5. 3. Wooden layers subject to four-point bending along the transversal direction. As we are interested in measuring the Young’s moduli in both longitudinal and transversal directions, we also performed a experiment consisting of the same setup and geometry described above, but with the longitudinal direction of the wood fibres oriented in the short direction of the specimen, as shown in Fig. 3(b). Five specimens were tested here and the average transversal Young’s modulus was 443 MPa. 4. CLT panels subject to three-point bending in the strong direction. Eight CLT panels were tested, made up of three 4-cm-thick layers with a length of 75 cm and a width of 39 cm. The span length between supports was 60 cm. We note that the main motivation of choosing these small dimensions was to produce shear effects in the panel. ASTM D3737 [27] suggests a span-todepth ratio near 5 to produce a high percentage of shear failure at a large extent. The outer layers were made up of timber pieces oriented in the strong direction of the panel. The central layer was made up of members oriented in the weak direction.

Fig. 3. Schematic representation of two four-point bending tests on a 4-cm-thick wooden layer.

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Refer to Fig. 4 for details. We measured experimentally loaddisplacement curves and we obtained an average initial stiffness of 21,483 N/mm obtained from their (initially) linear portions (up to a vertical displacement of 5 mm). 5. Multi-scale model updating As we are interested in improving the predictions of our numerical model, we propose an optimisation strategy based on a GA technique [28] to calibrate the micromechanical parameters which are either not well-known, or susceptible to considerable variations when measured experimentally. Such parameters are considered to be random and are listed in Table 1 with their corresponding intervals of variation. We must note that the term multi-scale finite element model updating refers here to the process of ensuring that multi-scale analyses result in models that better reflect the experimentally measured data than the original models. In this paper, this process is accomplished via the GA-based optimisation technique, which is used to minimise the discrepancies found between the measured data and the multi-scale model predicted data. GAs are stochastic global search and optimisation methods. GAs mimic the metaphor of natural evolution by applying the principle of the survival of the fittest to produce successively better approximations to a solution [31–33]. They follow a population-based approach which allows the optimisation process to be parallelised and hence to reduce the computational time. GAs start with an initial population consisting of various individuals, and each individual represents a particular solution to the problem. The population evolves over generations to produce better solutions. A fitness value is assigned to every individual of the initial population through an objective function that assesses the performance of the individual in the problem domain. Then, individuals are selected based on their fitness index and cross-over between them is performed to generate new offspring. Finally, mutation of the new offspring is performed to ensure that the probability of searching any subspace of the problem is never zero. These above mentioned processes iterate until the optimum solution is achieved depending on the convergence criteria of the problem. We remark here that by restricting the reproduction of weak candidates, GAs eliminate not only that solution but also all of its descendants, making the algorithm likely to converge within few generations. This represents the main justification of using GA in our analyses. The optimisation process starts with the creation of a random set of micromechanical parameters which are the design variables listed in Table 1. Table 1 also lists the range of variation for each design variable. The boundaries of each interval are chosen based on previous works ([15,29,30] and Refs. therein). Thus the optimisation of such parameters is performed to update the multi-scale model, but constrained to deliver physically possible values and consistent with the available experimental data. The performance is estimated by the fitness function which is quantified as the sum of the relative errors in the density and in the longitudinal Young’s modulus. The relative error in the density is calculated as jdnum  dexp j=dexp , and the relative error in the longi-

Fig. 4. Schematic representation of a CLT plate subject to three-point bending.

Table 1 Summary of the random micromechanical parameters chosen for this study and their corresponding intervals of variation (from [15,29,30] and Refs. therein). Micromechanical parameter Degree of cellulose crystallinity, f cc (%) Volume fraction of cellulose, f c (%) Volume fraction of hemicellulose, f h (%) Length of cellulose crystallites, Lcc (nm) Radial dimension of early-wood fibres, Rew (lm) Tangential dimension of early-wood fibres, T ew (lm) Cell-wall thickness of early-wood fibres, tew (lm) Radial dimension of late-wood fibres, Rlw (lm) Tangential dimension of late-wood fibres, T lw (lm) Cell-wall thickness of late-wood fibres, t lw (lm) Cell angle, h (°) Microfibril angle, MFA (°) Volume fraction of early-wood, f ew (%)

Interval of variation 45–60 30–51 25–29 26.5–36.4 37–40 28–30 3.1–4.3 31–37 25–27 4.3–8 10–27.5 0–22 67–80

tudinal Young’s modulus is obtained as jEnum  Eexp j=Eexp . Here, dnum and dexp are the density computed numerically and the average density measured experimentally (that is, 506 kg=m3 ). Similarly, Enum and Eexp are the longitudinal Young’s modulus obtained numerically and the average longitudinal Young’s modulus measured experimentally (that is, 12,712 MPa). Refer to Section 4 for further information on the above experimental average values. The two terms of the fitness function are of equal importance in this problem and therefore summing the two terms is sufficient to capture the sensitivities. After investigating different convergence criteria, the number of individuals per generation was fixed to 20. This relatively small number was attributed to the long CPU times associated with our multi-scale simulations. Using the Parallel Computing Toolbox in Matlab and its GA [34], this optimisation problem ran on 8 different processors. This reduced the computational time significantly and allowed us to investigate different convergence criteria. The optimisation process was repeated 5 different times to ensure consistency of the outcomes and achieve the global optimum. A cross-over rate of 0.8, which is the default value in Matlab, was adopted. This tends to be the optimum value for typical engineering problems. Selection, cross-over and mutation operators are all executed to create a new generation starting from the best fit individuals of the previous one. Multiple runs of the same optimisation allows the GA to start with different, randomly selected, initial conditions, enabling the analysis a better chance to converge to a global optimum [35]. We emphasise here that the GA is not proposed as the best optimisation method nor are the results accepted as the global optimum solution. The GA is used to tune the micromechanical parameters shown in Table 1 based on the experimental information coming from the density and longitudinal Young’s modulus, in order to improve our numerical predictions. The criterion adopted to stop the optimisation process is to set a tolerance of 0.05 to the average change in the value of the fitness function. Fig. 5 shows the variation of the best and average fitness

Fig. 5. Best and average fitness during optimisation.

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Table 2 Comparison between average experimental values and numerical results obtained from the present updated multi-scale model. Variable

Experimental value

Numerical value

12,712

12,842

506 443 21,921

497 447 21,062

Longitudinal Young’s modulus (MPa) Density (kg=m3 ) Transversal Young’s modulus (MPa) Stiffness of CLT plate (kg=cm)

values as the number of generations increases. The optimisation is terminated at the 51th generation, when the convergence criterion is achieved with a best fitness of almost 2.8%. The chromosomes of the best fit individuals are selected to update the micromechanical parameters. Thus, our multi-scale model provides an updated macroscopic mechanical response whose validation and numerical predictions are studied in the next Section 6. 6. Numerical simulations In this section we compare our numerical predictions with the experimental data and we perform additional simulations in order to investigate the structural response. Table 2 shows the comparison between experimental results and their numerical counterparts obtained from the present model. As expected, a very small difference is found between the numerical and experimental densities, with a relative error of 1.8%. Similarly, a small relative error of 1% is found for the values of the longitudinal Young’s modulus. As commented in Section 5, the present model was tuned to minimise the relative difference between the experimental and numerical values of the density and longitudinal Young’s modulus. We note that in our simulations, the finite element mesh had 35,380 nodes and 27,600 elements, as shown in Fig. 2(e). Furthermore, our model delivers a relative difference of only 0.9% between the numerical and experimental values for the transversal Young’s modulus. For the computational modelling of this test, we adopted the same finite element mesh shown in Fig. 2(e), but with the in-plane material axes rotated in 90°. CLT panels subject to three-point bending in the strong direction were modelled with a mesh discretised into 31,720 nodes and 28,080 elements as shown in Fig. 2(f). The central layer of the panel was made of the same homogenised material adopted for the modelling of the outer layers but rotated in 90°. Here, we computed the stiffness as the quotient between the applied load and the vertical displacement in the middle of the panel. We note that due to the small span-to-depth ratio of the CLT specimen

(equal to 5), shear deformations are expected to significantly contribute to the vertical displacement, and therefore to the CLT stiffness. Nevertheless, our numerical results are in good agreement with the experimental data, showing a relative difference of only 3.9%. The good agreement observed between our computational simulations and the experimental results gives us confidence to apply our model for further investigation on the mechanical properties of CLT. Thus, we study the behaviour of the sliding (or longitudinal) shear modulus (Go ) as a function of wood density. Refer to Fig. 6 (a) for details. Here, Go increases linearly from 254 MPa to 681 MPa as the density increases from 310 kg=m3 to 747 kg=m3 . This represents an increment in Go of almost 2.7 times. The modelling (not shown here) consisted of applying a linearly-increasing displacement field in a 4-cm-thick parallelepiped-shaped specimen. The displacements were applied in the longitudinal direction of wood fibres, producing angular distortion (uniform shear strain) in the plane formed by the longitudinal axis and the thickness direction. We measured the applied shear strain and the resulting shear stress, and with this information at hand, we computed the sliding shear modulus. To plot Fig. 6 (a) (and the following figures), we varied incrementally the cellwall thickness (and therefore the density), while the remaining micromechanical parameters were kept fixed. Refer to [15] for further details. Fig. 6(b) shows the variation of the rolling (or transversal) shear modulus (Gr ) as the wood density increases. To calculate Gr , we performed the same simulation described above but with the displacements applied perpendicularly to the wood fibre direction. We can observe here that by increasing the density from 310 kg=m3 to 747 kg=m3 , the value of Gr increases parabolically from 13.6 MPa to 178 MPa, representing an increment of almost 13 times the initial value. This shows how much sensitive Gr is to changes in density. Furthermore, we note that for a density of 506 kg=m3 ; Gr takes the value of 58.4 MPa, which is very close to the (constant) value

Table 3 Variation of the quotient Go =Gr as the wood density increases.

600 500 400 300 200 300

Go =Gr

310 405 497 584 668 747

18.68 11.16 7.66 5.76 4.60 3.83

200

Rolling shear modulus, MPa

Sliding shear modulus, MPa

700

Density (kg=m3 )

400

500

600

700 3

Densidad, kg/m

(a) Longitudinal direction

800

150

100

50

0 300

400

500

600

700

Densidad, kg/m3

(b) Transversal direction

Fig. 6. Rolling and sliding shear moduli as functions of wood density.

800

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E.I. Saavedra Flores et al. / Computers and Structures 177 (2016) 83–90 7

x 10

10

Effective shear stiffness, kg

Effective bending stiffness, kg⋅cm

2

9

1.5

1

0.5 300

400

500

600

700

800

x 10

8 6 4 2 0 300

400

500

600

700

800

3

3

Densidad, kg/m

Densidad, kg/m

(a)

(b)

Fig. 7. Effective bending (EIeff ) and shear (GAeff ) stiffness values in the strong direction of the CLT specimen as functions of wood density.

of 50 MPa recommended by the American standard ANSI/APA PRG 320 [36]. As Gr has proved to be very sensitive to density changes, we find it convenient to suggest the parabolic fitting curve obtained from Fig. 6(b) for design purposes. That is, Gr ¼ c2 =1638:81  0:2714c þ 39:5, with c the density measured in kg=m3 , and Gr the rolling shear modulus in MPa. This standard [36] also suggests to adopt a quotient Go =Gr ¼ 10 for the design of CLT. From our computational simulations, we obtain a quotient Go =Gr ¼ 7:5 (for the density of 506 kg=m3 ), which is also near the design value. As the quotient Go =Gr can be relevant in the design of CLT, we find it interesting to show in Table 3 the variation of this rate as the wood density increases. Here, we can observe that for densities close to 430 kg=m3 , our computed quotient Go =Gr approaches the recommended design value of 10. However, for heavier or lighter species, Go =Gr may become very different. Fig. 7 shows the variation of the effective bending (EIeff ) and shear (GAeff ) stiffness values in a CLT specimen as the wood density increases. The computational model (not shown here) consisted of a 12-cm-thick CLT plate, made up of three 4-cm-thick layers, similar to that shown in Fig. 2(f), but with a length and width of 2.4 m and 1.2 m, respectively. These dimensions were chosen in order to investigate the structural response of full-size CLT plates. Bending and shear stiffnesses were analysed along the strong direction of the CLT specimen, parallel to the long side of the panel. To plot Fig. 7(a), we considered the CLT panel subject to fourpoint bending (in order to exclude shear effects in the central portion of the element). The effective bending stiffness EIeff was computed directly from the quotient between the applied load and the vertical displacement in the middle of the panel, and by means of the standard formula of structural mechanics for the vertical deflection. The variation shown in Fig. 7(a) follows a linearly-increasing trend from 5.03e8 kg cm2 to 1.46e9 kg cm2 as the density varies from 310 kg=m3 to 747 kg=m3 . This represents an increment of 2.9 times the initial value of 5.03e8 kg cm2 . To compute the effective shear stiffness GAeff , we applied a uniform displacement field along the strong direction, directly on the upper face of the CLT panel and then we measured the resulting shear reaction in the same direction. The value of GAeff is calculated as the reaction force divided by the effective shear strain, which is computed as the applied displacement (on the upper face) divided by the thickness of 12 cm. Fig. 7(b) shows the variation of GAeff . The observed trend is parabolically-increasing from 1.07e7 kg to 9.38e7 kg, which represents an increment of almost 10 times the initial value. This result

emphasises the fact that the rolling shear modulus (which significantly controls the deformation mechanism in the central layer) is highly sensitive to density changes. 7. Conclusion In this paper we have investigated the mechanical properties of CLT by means of a homogenisation-based four-scale model. The material scales considered in our analyses are represented by the wood cell-wall, the wood fibres, growth rings and the macroscopic or structural scale. In addition, as wood shows a large amount of uncertainty in its properties, we have proposed an optimisation strategy based on a GA technique to tune those micromechanical parameters which are either not well-known or susceptible to considerable variations. The updating process has been made with experimental values of density and longitudinal Young’s modulus. Our numerical predictions have been validated successfully with relative errors smaller than 0.9% for the transversal Young’s modulus, and 3.9% for the stiffness of CLT plates subject to bending and shear effects. Furthermore, we have found that the sliding shear modulus varies linearly with density, with an increment of almost 2.7 times when the wood density increases from 310 kg=m3 to 747 kg=m3 . We have also found that the rolling shear modulus is much more sensitive to density changes, with an increment of almost 13 times when the density increases within the same range. We have also observed that the effective bending and shear stiffnesses increase by a factor of 2.9 and 10, respectively. The predicted quotient Go =Gr approaches the recommended design value of 10 for density values close to 430 kg=m3 . However, for heavier or lighter species, Go =Gr may become very different. We believe that the present modelling strategy represents a robust platform for further investigation on the structural response of CLT structures with a particular view to elaborate more accurate guidelines for structural design. This will be the subject of future works. Acknowledgments E.I. Saavedra Flores acknowledges the support from the Chilean National Commission for Scientific and Technological Research (CONICYT), FONDECYT research project No 1140245. References [1] Currilen Cavallari SA, Hermosilla de la Fuente MA. Modelación de paneles de madera contralaminada de pino radiata mediante elementos finitos multi-

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