Humidity fluctuations are a leading cause of damage in wooden constructions. In the case of glulam products, the multitude of possible layups concerning pith locations, diverse material properties across wood species, and the high computational cost associated with multi-field analysis have constrained many research efforts to focus on one specific glulam layup, consequently limiting the generalizability of the findings. To address this challenge, Monte Carlo simulations were employed to assess the significance of various factors. Based on which, two levels of simplification are proposed. The first level reduces the multi-layer problem to a single-layer one by applying appropriate boundary conditions. It substantially reduces the simulation costs and consequently facilitates sophisticated damage analysis, revealing the varying damage pattern across different board types. The second level of simplification further reduces the problem to a single-element model, enabling an analytical estimation of moisture stress. This level of simplification elucidates how factors such as moisture difference, material rotational angle, and other material properties influence the moisture-induced stress. Most importantly, it facilitates a rapid estimation of the critical moisture fluctuation range and the preferred sawing location of boards for different wood species, which can provide guidance to the production of higher moisture resistant glulam.
Hinweise
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Damage variable (for tension/shear) in direction i
\(E_i\)
Young’s modulus in direction i (MPa)
\(f_i\)
Strength in direction i (MPa)
\(F_{t/v, i}\)
Damage initiation criteria (for tension/shear) in direction i
\(G_{ij}\)
Shear modulus (MPa)
J
Moisture flux across the surface (\(\hbox {g} \; \hbox {s}^{-1} \; \hbox {mm}^{-2}\))
\(LOC_i\)
Location i to surface, i = 1, 2, 3..., illustrated in Fig. 5
\(S_u\)
Surface emission coefficient (mm/s)
t
Time (s)
u
Moisture content (−)
\(u_{EMC}\)
Equilibrium moisture content (−)
D
Diffusion coefficient (mm\(^2/s\))
Introduction
Wood is increasingly favored in construction for its high strength-to-weight ratio, renewability, and positive environmental impact. However, natural defects and size limitations inherent to wood necessitate the production of engineered wood products like glued laminated timber (glulam). Despite these advancements, moisture sensitivity remains a significant challenge in timber structures. Evaluation of 245 large-span timber structures revealed that moisture content variation contributes to 46% of structural damage causes (Dietsch et al. 2015).
To address these challenges, finite element method serves as a powerful tool for simulating moisture-induced stresses and assessing the response of wood products in diverse environmental conditions. Moisture content variation impacts wood in numerous ways. First, orthotropic hydro-expansion due to moisture changes can cause shape instability during technical drying and induce shrinkage stresses throughout both the drying process and the service life of timber structures (Ormarsson et al. 1998; Florisson et al. 2019; Brandstätter et al. 2023; Fortino et al. 2019). Additionally, moisture variation triggers mechano-sorptive creep and alters material parameters such as stiffness (Hassani 2015). An overview of the development of multi-field analyses that consider both hygroscopic and mechanical aspects of wood can be found in Yu et al. (2022 and Florisson and Gamstedt (2023). The constitutive model has been continuously improved and customized, aiming to cover diffusivity, hydro-expansion, elasticity, viscoelastic, mechano-sorption and damage/failure across different wood species.
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However, works studying the growth-ring lay-up effect in glulam remain limited. According to the DIC (digital image correlation) measurements by Lee et al. (2019), the moisture-induced strain in glulam along the width direction is found to be dependent on the characteristics of all layers due to the effect of adjacent laminas along the glue line. Autengruber et al. (2021) also revealed how different pith locations can lead to varying shrinkage crack patterns in single boards. Brandstätter et al. (2023) further investigated shrinkage cracks in glulam cross-section under different moisture drying conditions using the XFEM method. This research (Brandstätter et al. 2023), as well as many other works (Jönsson and Thelandersson 2003; Zhou et al. 2010; Ormarsson and Gíslason 2016) on glulam, is restricted to only one specific lay-up, and the generalizability of their findings to different lay-up types is not discussed. The major challenge in studying the growth-ring effect on glulam lies in the difficulty of covering the infinite possibilities of the lamina lay-up options. Franke et al. (2016), as well as Aicher and Dill-Langer (2005), conducted numerical studies using several lay-up cases and confirmed the significant influence of the cross-section lay-up on the stress distributions and their maximum values. The attempt to cover the large number of lay-up possibilities can be found in the stochastic study by Yu et al. (2023), where a Monte Carlo analysis, taking the pith-location in different layers as input uncertainty, is conducted. A stress concentration factor of up to 1.56 was identified in curved glulam beams caused by bending loads.
In this work, to determine the growth-ring effect on moisture-induced stress, a Monte Carlo analysis following the approach in Yu et al. (2023) is first conducted in sections “Material and method” and “Results and discussion”. The analysis identified the major influence of the growth-ring of the target layer and the minor influence from the neighboring layers. These findings then facilitate two levels of simplification methods in section “Development of simplification methods” to analyze moisture-induced stress more efficiently. The first level aims to accelerate numerical simulations for more complex analyses, while the second level employs stronger assumptions to analytically estimate shrinkage stress for different pith types, moisture conditions, and wood species. In section “Application”, using these two simplification methods, the damage patterns of different board types are presented, along with a fast way to identify critical moisture variation magnitudes and preferable board types.
Material and method
Finite element model
The numerical model is built using Abaqus. Sequentially coupled thermal-mechanical models are adopted. In the thermal model, according to the mathematical analogy between heat transfer and moisture diffusion (Fortino et al. 2009), the transient moisture field can be computed. The computed transient moisture field is then taken as field input for the mechanical model where the moisture induced stresses are simulated.
This study will conduct Monte Carlo simulations across four different wood species, involving a large number of parameters and significant variability. Therefore, the selection of an appropriate constitutive material model is critical and must satisfy two key criteria: (1) demonstrated capability to capture the moisture transfer and elastic-damage behavior of wood, and (2) availability of relevant parameters in the literature. Accordingly, the constitutive models for heat transfer and mechanical analysis are adopted from Yu et al. (2022) and Seeber et al. (2024). The principal equations used are outlined below.The values of the involved parameters for different wood species are listed in Tables 1, 2 and 3.
where \(\rho\), \(u\), \({\textbf {f}}\), \({\textbf {D}}\), and \(t\) are the density of the material, the moisture content, the flux vector, the diffusivity tensor, and time, respectively.
where \(\varvec{\epsilon }^e\), \(\varvec{C}^e\), and \(\varvec{\sigma }\) are the elastic strain, the elastic stiffness matrix, and the stress tensor, respectively.
where \(d_i\) is the damage variable, \(\alpha = 0.99\) and \(\beta = 0.1\) according to Seeber et al. (2024).
Figure 1 shows the overview of the finite element model. The simulated glulam beams consist of \(n\) (\(n = 5, 9, 11, 15, 19\)) layers of board with a cross-section of \(40 \, \textrm{mm} \times 200 \, \textrm{mm}\). Each board is assigned a cylindrical coordinate system, characterized by a pith coordinate \((y, z)\) using the center point as the origin (see Fig. 3). Although 2D model would be sufficient for the analysis in this work, the 3D eight-node brick element (C3D8) is chosen here to facilitate future extensions such as inclusion of slope of grain or analysis of curved glulam beams. In longitudinal direction, the model dimension was chosen with one element (4 mm) to model plain-strain conditions. Hence, after convergence analysis, each layer is discretized into 500 elements.
Fig. 1
Finite element model
×
The initial moisture content, \(u_{\textrm{EMC}}^{\textrm{initial}}\), is assigned to the entire model. Film conditions are then applied to the four side surfaces of the glulam, with a sink value of \(u_{\textrm{EMC}}^{\textrm{initial}} - \Delta u_{\textrm{EMC}}\). Figure 2 shows the equilibrium moisture content (\(u_{\textrm{EMC}}\)), calculated using the isotherms equation (Eq. 8) (Avramidis 1989). The temperature and relative humidity data are sourced from the Deutscher Wetterdienst for Munich (Germany), recorded over 365 days starting from February 2023 (Fig. 2). As a result, \(u_{\textrm{EMC}}\) fluctuates between 0.04 and 0.28. Since \(u_{\textrm{EMC}}^{\textrm{initial}}\) under standard storage conditions (T = \(20^{\circ }\textrm{C}\), RH = 65%) equals 0.12, the \(\Delta u_{\textrm{EMC}}\) (for the drying scenario only) is selected to range between 0.01 and 0.07.
Considering the measuring accuracy, all numbers are rounded to 2 scientific digits; Poisson’s ratios (\(\nu _{RL}\), \(\nu _{TL}\), \(\nu _{RT}\)) are calculated from the original literature assuming \(\frac{\nu _{ij}}{E_i} = \frac{\nu _{ji}}{E_j}\); for Beech and Spruce, moisture dependent data is used according to Hassani et al. (2015), the value listed here are at moisture content u = 0.12
Table 2
Hydro-expansion coefficient (Hassani et al. 2015; Fortino et al. 2009; Qiu 2015)
For larch, Qiu (2015) provided strength perpendicular to grain \(f_{t,90}\) = 2.31 MPa, which is adopted here for both \(f_{t,R}\) and \(f_{t,T}\)
×
Monte Carlo simulation
To analyze the growth-ring effect on moisture-induced stress, two test sets are conducted:
Test Set I 30 subsets are conducted for Test Set I. In each subset, the pith location of the middle layer are controlled to be the same. The 30 subsets corresponds to 30 types of middle layer, where the pith location are selected from a uniform grid as shown in Fig. 3, where \(y = -20, -60, -100, -140, -180\) mm and \(z = -200, -150, -100, -50, -25, 0\) mm. For each subset, 50 simulations are conducted. In each simulation, the pith location of all other layers varies in the way that the y and zcoordinate of the pith are uniform random variables in the range of \(y \in [-200, 0]\) mm and \(z \in [-200, 200]\) mm.
Test Set II For each simulation, each board layer is assigned a different random pith location. 300 simulations are conducted.
Fig. 3
The pith coordinates (y,z) are negative in the above examples as the individual board centers are taken as the origins. Pith location grid with 3 board example
×
Moreover, both Test Sets are conducted for different wood species in Table 1, for glulam that is composed of different number of layers n (n = 5, 9, 11, 15, 19), as well as for different humidity conditions (\(\Delta u_{EMC}\) = 0.01, 0.03, 0.05, 0.07).
Results and discussion
Overview of stress development
Figure 4 shows the spatial moisture and stress distribution in a simulation example at day 33 after placing into dry condition. The model parameters are:
number of layers: n = 11;
controlled pith of the middle layer: (y,z) = (− 60, − 150) mm;
humidity condition: \(\Delta u_{EMC} = 0.01\).
Fig. 4
Moisture and stress distribution at \(33^\textrm{rd}\) day after placing in dry condition of \(\Delta u_{EMC} = 0.01\)
×
It can be observed that as moisture diffuses out from the four surfaces, the highest tensile stresses predominantly develop in the surface regions.In contrast, in the case of moisture uptake, tensile stresses would form in the central region, with a lower magnitude compared to the stress concentration in the surface region observed in the drying cases. Moreover, the varying pith locations of each layer result in differences in the distribution of both moisture content and stresses, with particularly notable variations in stress distribution, highlighting the importance of analysing the growth-ring effect.
Figure 5 shows the moisture (\(\Delta u\)) and tangential stress (\(\sigma _T\)) development at the surface region of the middle layer over 90 days. For all locations and moisture conditions (\(\Delta u_{EMC}\)) studied here, the evolution of stress closely follows the pattern of moisture content change (\(\Delta u\)). However, while \(\Delta u\) continues to increase at a decreasing rate, \(\sigma _T\) begins to decrease after reaching a maximum. This occurs because the moisture content in the center region also decreases over time, albeit with a delay compared to the surface region. This delay results in a smaller moisture difference between the surface and center regions, leading to a decrease in the tensile stress in the surface region. Additionally, locations closer to the surface exhibit higher maximum stress and a shorter time to reach that maximum.
Fig. 5
Moisture and stress development at surface region (LOC i represents the outermost node of the \(i^\textrm{th}\) element from the surface)
×
Results of Test Set I
Figure 6 depicts the stress distribution within the same middle layer, considering various pith locations of the neighboring layers. Despite minor variations, the stress distribution in the middle layer shows remarkable similarity across different configurations of the neighboring layers.
Fig. 6
Tangential stress of the same middle layer with different layups to other layers: \(\Delta u_{EMC} = 0.01\), Pith of middle layer = (− 60, −150) mm, Time = \(1^\textrm{st}\) day
×
To provide a quantitative analysis, Fig. 7 presents the statistical distribution (obtained from 50 simulations of each type of middle layer) of tangential stress at four board corners (marked with x) for three exemplary types of the middle layer: with pith locations at (− 20, 0) mm, (− 40, − 50) mm, and (− 60, − 150) mm, respectively. The average stress across different locations of these three types ranges from 0.79 to 1.69 MPa. In contrast, the stress variation within a specific corner of a particular middle layer is confined to a range smaller than 0.3 MPa.
Fig. 7
Statistic distribution of tangential stress at the corners of the middle layer (\(\Delta u_{EMC} = 0.01\), Time = \(1^\textrm{st}\) day)
×
A normal distribution is employed to fit the data, and the standard deviations consistently measure below 0.07 MPa. This observation suggests that the influence of the layups of neighboring layers on stress distribution is limited.
Furthermore, it is observed that points sharing the same material rotational angle \(\theta\) exhibit consistent average stresses, regardless of the middle layer type, such as Point C of the first and second example in Fig. 7. To validate this observation, stress values are plotted against the rotational angle for all simulations across all 30 types of middle layers in Fig. 8.
Fig. 8
Stress vs rotational angle: \(\Delta u_{EMC} = 0.01\), Time = \(1^\textrm{st}\) day, LOC = 1 (LOC definition see Fig. 5)
×
Based on the above analysis, the following conclusions can be drawn regarding the stresses in the middle layer:
Observation I: stresses are minimally affected by the layups of neighboring layers.
Observation II: stresses exhibit a strong correlation with the rotational angle of the material.
Results of Test Set II
In this test set, the goal is to validate the aforementioned features across various wood layers, times, locations, and wood species.
Figure 9 displays the tangential stress values versus rotational angles in different layers of a 9-layer glulam beam. It can be seen that, except for the top and bottom layers, all layers exhibit an almost identical relation between stress and rotational angle. The top and bottom layers, however, show two distinct types of curves. This is due to the boundary conditions at locations C and D for the top layer and at locations A and B for the bottom layer, which are essentially free.
Fig. 9
Stress vs rotational angle at different layers: species = Beech, \(\Delta u_{EMC} = 0.03\), Time = \(33^\textrm{rd}\) day, LOC = 3
×
As shown in Fig. 10, one day after placing the beam in dry conditions, tangential stress is only evident at location Loc = 1 (location definition is shown in Fig. 5) and remains at 0 MPa at all other locations. After one month, stress variations start to appear at all locations but are smaller further away from the surface. This can be explained by moisture penetrating deeper into the beam over time. At each time point, the further the distance from the surface, the smaller the magnitude of the moisture content change.
Fig. 10
Stress vs rotational angle at different locations and time points: species = Beech, \(\Delta u_{EMC} = 0.03\)
×
It is observed that the variation of stress at the same angle becomes larger over time. At three years after placing the beam in dry conditions, the correlation is almost not noticeable.
Hence, it can be concluded that the aforementioned features found in Test Set I are valid for:
all board layers except for the top and bottom;
different locations where the distance to the side surface is between (0, Y) mm.
a limited time period between (0, X) days.
as observed in Monte Carlo simulations, for a board dimension of 40 mm \(\times\) 200 mm, Y is no less than 30 mm and X is no less than 30 days.
Development of simplification methods
First level of simplification
Based on the Observation I from Test Set I and Test Set II, where the stress in most board layers is only slightly influenced by the neighboring layers, it is logical to assume that the multi-layered glulam problem can be simplified to a single-board problem. This simplification is depicted in Fig. 11, where the influence of the neighboring layers is abstracted into a rotational boundary condition.
Fig. 11
First level of simplification
×
To validate this assumption and to understand why the influence of neighboring layers is minimal, it is crucial to comprehend the underlying mechanisms of stress development.
From the spatial distribution of moisture content and stress shown in Fig. 4, it can be seen that after one month, only a very limited depth is influenced by the humidity change due to the slow diffusion process in wood. Consequently, most of the central region exhibits almost zero stress, while only the surface region experiences high tensile stress due to the shrinkage in this area. Based on this observation, the central region can be considered a rigid body, maintaining almost constant moisture content and stress over a certain time range. Therefore, the problem can be represented using the sketch shown in Fig. 12. The steps in Fig. 12 and the corresponding moisture and mechanical states are:
Fig. 12
Sketch of the deformation at three states
×
Initial State
Moisture state\(u_P^{\text {initial}} = u_{\textrm{EMC}}^{\textrm{initial}}\) for any point P in Center Region and Surface Region.
Strain state Zero deformation in Center Region and Surface Region.
Stress state Zero stress in Center Region and Surface Region.
Virtual Free-Shrinkage State
This is a virtual state with assumed free boundary condition Surface Region can shrink freely without any restriction from the Center Region or neighboring layers (i.e., no connection between the Surface Region and the Center Region, and no connection between layers).
Moisture state\(u_{P}^{\text {free}} \in [ u_{\textrm{EMC}}^{\textrm{initial}} - \Delta u_{\textrm{EMC}}, u_{\textrm{EMC}}^{\textrm{initial}}]\) point P in the Surface Region, and \(u_{\text {Center}}^{\text {free}} \approx u_{\textrm{EMC}}^{\textrm{initial}}\) for the Center Region.
Strain state Free shrinkage for the Surface Region according to \(u_{P}^{\text {free}}\); approximate zero deformation for the Center Region.
Stress state Almost zero stress in all regions due to the free shrinkage.
End State
Boundary condition Deformation of the Surface Region is restricted by the Center Region and the Surface Regions of two neighboring layers.
Moisture state\(u_{P}^{\text {end}} \in [ u_{\textrm{EMC}}^{\textrm{initial}} - \Delta u_{\textrm{EMC}}, u_{\textrm{EMC}}^{\textrm{initial}}]\) for the Surface Region, and \(u_{\text {Center}}^{\text {end}} \approx u_{\textrm{EMC}}^{\textrm{initial}}\) for the Center Region.
Strain state Approximate zero deformation for the Center Region. Deformation in the Y direction of the Surface Region recovers to approximate the same state as the Center Region, with slight differences due to variations in global stiffness between layers.
Stress state Approximate zero stress for the Center Region; tensile stress for the Surface Region due to the strain difference between the End State and the Free-Shrinkage State.
From the above analysis, it is clear that stress arises because that the surface region is restricted in the End State, in comparison to the Virtual Free-Shrinkage State. This restriction is caused by the two neighboring layers. Moreover, how strong the restriction is dependent on the local stiffness of the neighboring layer at the surface region, which can be determined by the local material direction.
If the material direction of the analyzed layer aligns with its two neighboring layers, then the surface region at the end state should return to the same level as the initial state, resembling the boundary condition depicted in Fig. 11. However, due to differences in the pith location between layers, the returned level is not exactly the same as the initial state, which explains the fluctuation of stress within the same layer as observed in Fig. 7 in Test Set I.
Nonetheless, because the surface region has a shallow depth, the center region can be considered almost as a rigid body. This means the recovered level of the surface region is nearly equivalent to the initial state, and the influence of the neighboring layers is small compared to the influence of the center region, explain the Observation I in Test Set I.
Consequently, Fig. 13 presents the stress-angle relationship as simulated by both the full glulam analysis (real case) and the first level of simplification method for different species. The strong quantitative agreement between these results validates the simplification method across different locations and various wood species.
Fig. 13
Stress versus rotational angle of different species at different locations (from inner 5 layers of 9-layer glulam): \(\Delta u_{EMC} = 0.03\), Time = \(14^\textrm{th}\) day
×
To summarize, the first level of simplification relies on the observation that varying the pith location of neighboring layers has minimal impact on the stress levels of the analyzed layer. Therefore, the primary assumption and limitation of this simplification is the disregard of this influence. Additional, such simplification is not valid for the most top and bottom layer.
The main advantage is that it reduces the multi-layer problem to a single-layer problem, making it easier to perform more complex analyses that requires sophisticated material models and higher computational cost. An example is the damage evolution analysis that will be shown in the next chapter.
Second level of simplification
Although the first level of simplification significantly reduces computational costs, certain observations from the Monte Carlo simulation (as shown in Fig. 13) remain unexplained:
Why does the stress in the surface region exhibit such a strong non-linear correlation with the rotation angle?
Why does this correlation vary between different species?
What is the primary material parameter that determines the maximum stress?
If we can derive an analytical equation to explain this non-linear correlation, we could generate the correlation curve and estimate the maximum stress for different species in a matter of seconds, instead of conducting extensive numerical simulations for various types of boards and species.
To achieve this, the problem needs to be further simplified to the single-element level, as shown in Fig. 14. Building upon the problem abstraction outlined in Fig. 12 and the success of simplifying the influence of the neighboring layers to rotational boundary conditions, it is logical to assume that the influence of the center region can similarly be characterized by a fixed boundary condition on one side.
Fig. 14
Second level of simplification
×
Neglecting the effect in the longitudinal direction, the stiffness matrix (\(\varvec{C}\)) and rotational matrices (\(\varvec{T^{\epsilon }}\), \(\varvec{T^{\sigma }}\)) are given by:
If the element P can shrink freely (as shown in Fig. 12 Free Shrinkage State), the free shrinkage strain in the local (\(\varvec{\epsilon }^{free}_{P,local}\)) and global coordinate system (\(\varvec{\epsilon }^{free}_{P,global}\)) are:
where \(\alpha _R\) and \(\alpha _T\) are the hydro-expansion coefficients. \(u^{end}_P\), \(u^{free}_P\), \(u^{initial}_P\) represent the moisture content of element P at the End, Virtual Free-Shrinkage, and Initial State, respectively.
Hence, the strain changes \(\Delta \varvec{\epsilon }_{P,global}\) between End State and Free Shrinkage State can be calculated as:
additionally, assume the element in the Center Region next to element P to have the same rotational angle as element P, the strain at the End State in Y direction of center region (\(\epsilon ^{end}_{{Center}_{\ Y}}\)) becomes:
where \(C''_{ij}\) are the components of \(\varvec{C} \varvec{T}^{\epsilon -1}\).
The validation of the above solutions is presented in Fig. 13 for different locations and wood species. Moreover, it is important to note that both simplification methods have also been validated for various \(\Delta u_{EMC}\) conditions, different time (ranging from \(1^{st}\) to \(14^\textrm{th}\) day), and different stress components (\(\sigma _R\), \(\sigma _T\), \(\tau _{RT}\)).
To summarize, the second level of simplification further abstracts the influence of the center region on Element P of the surface region as a fixed boundary condition, which is mathematically represented by Eqs. 16–19. Correspondingly, this method is valid only within the elastic range and during periods when moisture changes in the center region are minimal.
The major advantage is the clear elucidation of how stress is influenced by the rotational angle, stiffness, and hydro-expansion coefficients. This means that for any new wood species, a quick estimation of the moisture-induced stress level can be achieved without conducting numerical simulations. It also provides information such as the critical rotational angle, critical moisture range, and preferred board type, as will be demonstrated in the next chapter.
Application
Application of second level of simplifcation
Critical rotational angle
According to Eqs. 24 and 25, the stress at two extreme angles can be calculated:
\(\sigma ^{end, 0^{\circ }}_{P_{T}}\) and \(\sigma ^{end, 90^{\circ }}_{P_{R}}\) represent approximately the maximum tangential stress and maximum radial stress, respectively. Accordingly, Fig. 15 shows examples of boards that will exhibit the highest tangential/radial stress.
Fig. 15
Examples of boards with locations exhibiting highest stress (highlighted in red) (color figure online)
×
The first board type in Fig. 15, where the pith is located at (− 20 mm, 0 mm), is frequently considered a critical case in various studies (Brandstätter et al. 2023; Jönsson and Thelandersson 2003; Zhou et al. 2010). However, this assumption appears to lack sufficient justification. While this board type does represent the scenario with the highest tangential stress induced by moisture, it remains unclear if it corresponds to the earliest initiation of damage and how extensively the damage will develop. These issues will be addressed in the following section.
Critical moisture fluctuation range
Instead of the absolute stress value, a more meaningful parameter to check is the damage variable. According to the initiation criterion of Separated Damage Mode (Seeber et al. 2023):
The damage variable versus the rotational angle can be seen in Fig. 16, according to the parameters from Tables 1, 2 and 3.
Fig. 16
Damage variable versus rotational angle (\(\Delta u_{EMC}\) = 0.03, LOC = 1, Time = \(14^\textrm{th}\) days)
×
Moreover, this method can also be adopted for different failure criteria. As an example, the multi-surface Tsai-Wu failure criteria is considered for spruce (Lukacevic et al. 2017) with the corresponding parameters shown in Table 4:
where \(\Delta u\) is moisture fluctuation magnitude, and \(\Delta u = (u^{end}_{\text {Center}} - u^{end}_{P})\).
Hence, for a given wood species, whether \(F_T^{max}\) is larger than \(F_R^{max}\)-and thus whether boards with a surface angle of \(0^{\circ }\) or \(90^{\circ }\) will exceed the damage criteria first-depends on the ratios \(\frac{E_T \alpha _T}{f_T}\) and \(\frac{E_R \alpha _R}{f_R}\). The assumption that the board types in Fig. 15a are most critical in terms of damage initiation holds true under the condition that \(\frac{E_T \alpha _T}{f_T}> \frac{E_R \alpha _R}{f_R}\).
In addition, the critical moisture fluctuation range \(\Delta u_{\text {critic}}\) can be calculated using:
$$\begin{aligned} \Delta u_{\text {critic}} = min (\Delta u |_{F_T^{max}=1}, \Delta u |_{F_R^{max}=1}) \approx min\left( \frac{f_T}{E_T\alpha _T}, \frac{f_R}{E_R\alpha _R}\right) \end{aligned}$$
(37)
This value represents the change in moisture content that would lead to a violation of the damage criteria. If the moisture change remains below this threshold, the boards can be considered safe from shrinkage cracks. Consequently, \(\Delta u_{\text {critic}}\) for beech, pine, and larch are 2.5%, 2.4%, and 0.8%, respectively. Similarly, \(\Delta u_{\text {critic}}\) for spruce can be calculated according to the multi-surface Tsai-Wu criteria and the result is 1.4%.
Based on the adopted failure criteria, the glulam made from beech and pine are less prone to the shrinkage damage caused by moisture changes. Their \(\Delta u_{\text {critic}}\) values align well with the allowed fluctuation range of 2%, as suggested in an early research document by van der Velden and Kuipers (1976). However, spruce and larch are more susceptible to moisture-induced damage, indicating a need for stricter control on moisture fluctuation.
Preferred board type
Similar to Eq. 37, the critical moisture fluctuation range for all rotational angle can be inversely calculated, and the results for beech and spruce are shown in Fig. 17.
Fig. 17
Critical moisture content change
×
It can be seen that, for beech wood, by selecting boards with rotational angles between \(30^{\circ }\) and \(150^{\circ }\), the critical moisture content for beech wood can be increased to 3.1%. For boards with rotational angles within the range of \(45^{\circ }\) to \(135^{\circ }\), the critical moisture content can reach as high as 4.7%. Similarly, for spruce, the critical moisture contents are 2.7% and 4.5% for the rotational angle ranges of [\(30^{\circ }, 150^{\circ }\)] and [\(37^{\circ }, 143^{\circ }\)], respectively.
It is essential to point out that the conclusion is based on the simulation results, where the fluctuation range of \(\Delta u_{EMC}\) is below 0.07, and the elasticity of wood is considered to be moisture-independent. Additionaly, this work considers mainly the material rotational angle in the cross-section but not any fiber deviation. In reality, the hardwood species such as beech always presents stronger curvature in the longitudinal direction, posing challenging in controlling the cutting angles of the boards. For such species, the plausible way to estimate the moisture induced stress could be employing the first level of simplification and additionally implementing the fiber deviation information, which can be obtained through the scanning technique (Seeber et al. 2023; Huber et al. 2021; Lukacevic et al. 2019).
Application of first level of simplification: damage development
Based on Observation I from the Monte Carlo analysis, it can be inferred that the development of moisture-induced damage in a board within a glulam structure is primarily influenced by the material properties of the board itself, rather than the material properties of the neighboring layers. Hence, employing the first level of simplification, it streamlines the analysis process while still providing reliable insights into the damage evolution. Figure 18 shows the damage patterns of 25 types of boards with varying pith directions. In this simulation, the material used is beech, and the \(\Delta u_{EMC}\) is 0.03.
Fig. 18
Damage parameter d for tangential direction (see Eq. 7) of boards with different pith coordinate (y, z)
×
It can be seen that additional to the damage at the surface regions, significant damage can evolve inside the board. As the damage variable shown here is the tangential one, which is the most major damage for beech wood, it advances mainly in the radial direction. These simulated damage patterns qualitatively resemble the drying cracks observed in the real glulam beams in the work of Franke et al. (2015) and Bucur (2011). However, quantitative validation is still necessary, particularly concerning damage depth, the \(\Delta u_{EMC}\) level that activates damage, and other factors. Moreover, as the simulation is a sequentially-coupled moisture-mechanical analysis, the evolution of damage is assumed to have no influence on moisture diffusion in this study.
Nevertheless, examining the evolution of moisture content, stress, and damage parameters at different simulation increments, as depicted in Fig. 19, provides a possible explanation for the major radial cracks: the onset of damage corresponds with changes in moisture content, starting from the surface region. Afterwards, due to the damage, the affected areas can withstand less stress, leading to stress redistribution inside the entire board. Given the low tensile strength in the tangential direction, significant tangential damage is then formed and evolves in the radial direction.
The damage pattern in different boards varies significantly due to the location of the pith. As shown in the green box in Fig. 18, for boards where the z-coordinate of the pith is near the surface region (i.e., pith = (y, z), with y \(\le\) − 60 mm, z \(\approx\) − 100 mm), the damage remains localized at the surface, since the penetration direction is almost vertical. Conversely, as highlighted in the red box, for boards where the y-coordinate is within the thickness of the board (i.e., pith = (y, z), − 20 \(\le\) y \(\le\) 0 mm), the damage progresses almost horizontally, penetrating the entire width of the board. In such cases, serious concerns can be raised about the stability of the entire glulam structure.
Fig. 19
Damage evolution of board with pith of (− 20 mm, 0 mm) after 36, 100, 104, 106 h
×
Conclusion
This study employed Monte Carlo simulations to analyze moisture-induced stress in glulam, where the pith locations of different layers are taken as the input random variables. Three key observations can be concluded from the Monte Carlo results:
1.
The moisture-induced stress varies significantly among boards with different pith locations, highlighting the importance of considering the growth-ring effect.
2.
The influence of neighboring board types on shrinkage stress is minimal, suggesting that focusing on features of the analyzed layer itself rather than neighboring layers is sufficient in many cases.
3.
A strong non-linear correlation exists between the moisture-induced stress and the material rotational angle in the surface region.
To address the need for reducing computational costs and understanding of how material parameters influence moisture-induced stress, two levels of simplification methods are proposed based on the observations. These methods yield the following insights:
1.
The nonlinear relationship between moisture-induced stress and material parameters can be approximated by mathematical equations;
2.
The common assumption - a board with pith at the middle of its bottom surface represents the most critical case - holds true if \(\frac{E_T \alpha _T}{f_T}> \frac{E_R \alpha _R}{f_R}\);
3.
Spruce and larch are more sensitive to moisture fluctuations compared to beech and pine, according to the material parameters provided in literature;
4.
In the analyzed example of beech, moisture-induced damage primarily advances in the radial direction.
5.
Although damage is initiated by the moisture-induced stress that is localized at the surface region, the advancement of damage can penetrate the entire width of the board, for boards where the height of pith is within the board’s thickness.
Declarations
Conflict of interest
The authors declare no Conflict of interest.
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