Macroscopic Model for Spatial Timber Plate Structures with Integral Mechanical Attachments
Publication: Journal of Structural Engineering
Volume 146, Issue 10
Abstract
The use of integral mechanical attachments (IMAs) in spatial free-form timber plate structures has been revitalized by the production of engineered boards and recent developments in digital fabrication and computer-aided design. Despite the widespread interest in utilizing such structural systems, there have been very few efforts to develop modeling strategies that can be used by practitioners. Detailed finite-element (FE) models are typically computationally expensive and require advanced expertise. Accordingly, this paper introduces a simplified yet robust macroscopic modeling technique for spatial free-form timber plate structures with IMAs. This approach employs only beam and spring elements to simulate the structural behavior. The macromodel was introduced and the associated mechanical properties were computed. FE models constructed using shell elements and the results from recent experiments performed on a full-scale prototype were used to verify the proposed macromodel. The results showed that the response of the macromodel was closely in line with that of the experiments and FE models. Despite its simplicity, the macromodel is robust and can simulate the behavior of integrally attached timber plates. It was also demonstrated that the computational time for the macromodel is significantly less than the FE simulation.
Introduction
This section provides an introduction to the integral mechanical attachment and its application in timber plate structures. The motivation and overview of the current study is also presented.
Integral Mechanical Attachment
The integral mechanical attachment (IMA), which is used to join timber elements with only wood–wood interlocking mechanisms (i.e., without connectors such as screws, nails, adhesives, and metal fasteners), is considered one of the oldest joinery techniques (Messler 2006; Robeller and Weinand 2015). In the last decade, the introduction of digital geometry processing to the design of complex freeform structures and the application of computer numerically controlled (CNC) machines for fabrication and robotic arms for assembly have resurrected the IMA technique. Moreover, engineered timber products such as laminated veneer lumber (LVL) planar boards have addressed the deficiencies of solid woods through their enhanced mechanical properties (Rezaei Rad et al. 2019a).
This paper focusses on two specific types of IMAs: multiple tab-and-slot dovetail joints [Fig. 1(a)], and mortise-and-tenon or through-tenon (TT) joints [Fig. 1(b)]. The flexural behavior of both types of IMAs was studied by Roche et al. (2015), Roche (2017), and Nguyen et al. (2019a). Rezaei Rad et al. (2019a, b) investigated the performance of TT joints under tension and in-plane (IP) (edgewise) loads, respectively. Additionally, the influence of dovetail and TT joints on the global behavior of timber folded plate surfaces was investigated by Stitic et al. (2018, 2019). They concluded that the semirigid behavior of IMAs considerably affects the performance of spatial timber plates and therefore should be considered in the design process.
Timber Plate Structures with IMAs
An interdisciplinary design framework that combines the knowledge of architects, engineers, and computer and robotic scientists is critical to the construction of novel timber plate structures. Therefore, the laboratory for timber construction (IBOIS) at École Polytechnique Fédérale de Lausanne (EPFL) recently designed and participated in the construction of the first timber folded plate theatre using only wood–wood IMAs. Gamerro et al. (2018) experimentally studied the flexural behavior of the IMA joints used in the structure and found that they can adequately satisfy the relevant design requirements.
The IMA technique has also been used to construct freeform surfaces. This is partly the result of a biologically inspired computational design strategy that was introduced by Magna et al. (2013) and Krieg et al. (2015a, b). The IMA finger joints, including pairs of parallel cross-screws, were used to achieve the desired mechanical properties. The framework addressed challenges in the fabrication, assembly, and service-life performance of the structure. With the support of IBOIS researchers, it was implemented in the design and construction of freeform integrally attached timber plate (IATP) structures using only wood–wood connections (Fig. 2). The lightweight construction system uses CNC routers to fabricate timber plates equipped with wood–wood connections along the plate edges. The computational framework used for prefabrication and assembly was described by Robeller et al. (2016) and the parts that are relevant to the current work were discussed further in the section in which the IATP system and assembly process are described.
Motivation and Overview of Present Study
The numerical models used to analyze plate structures typically lie somewhere along a spectrum of complexity. Finite-element (FE) models with three-dimensional (3D) solid elements, which can be viewed as existing on one end of that spectrum, are detailed, complex, and require a high level of sophistication. However, they do provide the benefit of enabling an evaluation of the state of stress across the plate thickness. In less complex models, the plate thickness is neglected, and two-dimensional (2D) shell elements can be used to simulate the behavior of the plates. Both of the aforementioned approaches provide enhanced simulation capabilities to capture stress–strain response and local failure modes. However, they are not without their challenges. First, a refined mesh discretization is required. Furthermore, a high level of expertise is needed to develop reliable models and to postprocess and interpret the results (Bao et al. 2008; Pantò et al. 2016). In other words, detailed FE models are not suitable for most industry applications (Bao and Kunnath 2010; Kareem and Pantò 2019) because of the associated computational expense, especially for large-scale structures (Caliò and Pantò 2014). Moreover, it has been shown that more-detailed models do not always provide improved results compared to simpler models in large-scale systems (Caliò et al. 2012; Pantò et al. 2016). On the other hand, more-simplified models, which utilize only beam and spring elements, can be used to simulate structural response. These models, which are generally referred to as macromodels, are computationally inexpensive and involve less complexity in their construction and analysis and in postprocessing the results (Pantò et al. 2019). The primary goal of the macromodeling approach is to simulate the global behavior of a structure (Pantò et al. 2018).
This research introduces a novel macromodeling approach for timber plate structures which can be implemented in structural engineering practice. The proposed macromodel is only valid for linear static analysis. Fig. 3 presents an overview of the contents of the paper. The mechanical characterization chapter describes the assembly of the IATP system, the kinematic degrees of freedom (DOFs) associated with timber plates and IMAs, the force flow, and equilibrium equations [Fig. 3(a)]. The macromodel is then introduced and the mechanical properties associated with the beam/spring elements are determined [Fig. 3(b)]. Next, the response of an individual plate and a full-scale subassembly is simulated using the macromodel and the results are verified using experimental data (only for the full-scale subassembly) and FE models using shell elements (for both the individual plate and the full-scale subassembly) [Fig. 3(c)]. Lastly, the conclusions of the study are discussed.
Mechanical Characterization of Integrally Attached Timber Plates
Description of the IATP System and Assembly Process
The IATP structural system consists of several planar elements with rectangular/parallelogram shapes, which are connected by TT and dovetail joints. In brief, the target surface of a freeform structure is transformed into a target geometry [Figs. 4(a and b)] and the configuration of the timber plates is determined [Fig. 4(c)]. Double-layered IATP systems include the assembly of multiple four-sided hexahedron-shaped boxes, denoted in Fig. 4(d), where . Each box consists of top and bottom plates [denoted and in Fig. 4(d), respectively] and two cross plates [denoted cross longitudinal, , and cross transverse, , in Fig. 4(d)]. Within each box, the tenons are located along the edges of the and plates, and their mates (slots) are located in the and plates. The and plates are connected via multiple dovetail-shaped joints. Each box is then connected to its neighbor with only the TT joints. Each timber box is labeled with a normal vector (, ). The collective set of normal vectors within a structure can be algorithmically determined so that they comply with the target surface to achieve the desired geometry.
A 1-degree-of-freedom process is used to assemble the plates (Robeller et al. 2016), meaning that there is only one translation vector in a 3D space to (dis)assemble adjacent plates; and accordingly, the joint geometry reduces the assembly degrees-of-freedom to that single vector. Two primary steps are involved in assembling an IATP system. In the first step, the plates are assembled to form each box (intrabox assembly). The and plates first are connected along vector . Fig. 4(d) shows being connected to along vector . Next, the and plates are connected simultaneously to the corresponding and plates along the internal assembly vector, [Fig. 4(d), ]. The neighboring boxes are assembled as part of the second step (interbox assembly). Each box is simultaneously connected to its two nearest neighbors along the same external assembly vector (). Fig. 4(d) shows being simultaneously connected to and along vector .
IATP Kinematics and Equilibrium Equations
Given the primary focus on the global (structural) scale, the overall behavior is deemed to be governed by the in-plane and out-of-plane (OP) response of the IATP elements and the behavior of the TT joints. To formulate the equations governing the IP and OP behavior, only the relative displacements associated with the perimeter of the plate are considered. In other words, only the DOFs associated with nodes located at the IATP perimeter are used. This reduces the size of the system of equations to be solved in the stiffness method, which in turn lowers the overall computational expense. The governing equations are presented in the next two subsections and in Appendices I–VI.
In-Plane Behavior of IATPs
Depending on the location of each timber box, the plate types (e.g., and or and ), and the geometry (parallelogram or rectangular), an IATP element can be subjected to three main IP load cases:
1.
An adjacent box applies tension forces to the box of interest [e.g., in Fig. 5(a)], which subjects plates and to tension forces. The applied loads (, , , and ) and the undeformed and deformed and plates are shown in Fig. 5(b). The IP loads are transferred from TT joints 1–5 to the and plates and exit TT joints 1′–5′.
2.
An adjacent box applies compression forces to the box of interest [e.g., Fig. 5(a)]. Due to the edgewise contact of the plates [e.g., in Fig. 4(d), and of have edgewise contact with of and with of ], the entire perimeter (edge) of the and plate transfers the load to and , and then to the next box. Similar to Case 1, Fig. 5(b) shows the undeformed and deformed shapes and the applied loads (, , , and ).
The kinematic formulation of the IP behavior of a single IATP element was adapted from (1) Christovasilis and Filiatrault (2013) and Casagrande et al. (2016), which focused on describing the in-plane behavior of sheathing panel elements in wood-frame shear walls; and (2) Pantò et al. (2018), which focused on characterizing the in-plane behavior of masonry vaults. Additional assumptions were made to ensure that the formulation is compatible with the mechanical constraints of the IATP system.
The equations governing the IP behavior of the and plates, the associated local and global coordinate systems, and the applied loads in Figs. 5(a and b), are formulated in this section and in Appendices I–IV. A similar formulation can be established for the and plates [Figs. 5(c and d)]. Five kinematic DOFs, including uniform shear () [Fig. 6(a)] and uniform flexural deformations () [Fig. 6(b)], two rigid-body translations ( and ) [Fig. 6(c)], and rigid-body rotation ()[Fig. 6(c)], are defined for each IATP element. The vector is used to describe these kinematic DOFs when the element responds within the linear elastic range. The corresponding force vector is also defined, where and are the IP forces along the global - and -axes, respectively, and is the IP shear force. The IP moment, , is decomposed into two components: which comes from the neighboring boxes and causes IP rigid-body rotations, and , which is the IP moment that causes flexural deformations.
The IP displacement of node , which represents the th IMA located at the perimeter of the IATP element (Fig. 6), is described by the vector in the global coordinate system (GCS) . The vector includes contributions from shear deformation , flexural deformation , rigid-body translation and rigid-body rotation . The principle of superposition is used to compute [Eq. (1)], which is deemed valid because only linear elastic responses are considered
(1)
To determine the location of node at each load–displacement step, a local coordinate system (LCS) and a coordinate vector, are defined. The latter specifies the position of node in the LCS with respect to its origin at each load step. The origin of the LCS is located at the center of the panel (Fig. 6) and, in the undeformed and undisplaced condition, is assumed to be parallel to the GCS. As noted previously, the IP deformation is computed as the vector sum of the deformations from the various sources. Consequently, is updated at each load step and used for the calculations in the next step. The detailed mathematical formulation, including the relationship between (1) the and vectors, and (2) the and vectors, is presented in Appendix I.
To compute the IP stiffness of an IATP element, a virtual displacement is associated with node where = matrix representing the partial derivative of IP displacements with respect to . The calculation needed to obtain each entry of the matrix is given in Appendix II.
(2)
Assuming that there are TT joints along the edge of the timber plate, the virtual work principle () is applied. The associated mathematical formulation is provided in Appendix III. The global force vector is expressed in terms of the internal forceswhere = global force vector; = matrix of partial derivatives of the IP deformation (Appendix II); = internal force vector of each IMA joint [Fig. 5(b)]; and , , and = axial force in global -direction, IP shear force, and bending moment, respectively (Appendix III).
(3)
Finally, the IP stiffness of the IATP, including the plate () and IMAs (), is (with additional details provided in Appendix IV)
(4a)
(4b)
In summary, the IP behavior of a timber plate equipped with TT IMAs can be simulated while accounting for the axial (), shear (), and flexural () stiffnesses of the timber plate and the semirigid behavior of the TT joints under tensile () and edgewise shear () loads. The calculation of these stiffnesses are discussed in the macromodel section.
Out-of-Plane Behavior of IATPs
Fig. 7 shows the loads applied to the box. Considering (1) the loads imposed by the neighboring boxes [ and in Fig. 4(d)] on the box, and (2) any external load applied directly to , an IATP element can be subjected to two main OP load cases:
1.
The and plates are subjected to OP loads from the neighboring boxes via the TT joints indexed 1 to 5 [Fig. 7(a)]. These loads, which are shown as , , and in Fig. 7(a), flow through the plate and exit from TT joints 1′–5′ to the and plates. The and plates resist most of the bending moment, and it will be shown that they contribute the most to OP behavior. Because the TT joints are located along the four edges, the and plates are subjected to biaxial bending. Fig. 7(b) shows the loads applied to the and plates, as well as the undeformed and deformed shapes.
2.
IP loads produced by the and plates impose OP loads on the and plates. IP loads from the and plates enter (1) the plate through TT joints 3′–5′, and (2) the plates through TT joints 1′ and 2′ [Fig. 7(a)]. Due to the location of the dovetail and TT joints [Fig. 7(c)], the and plates are subjected to uniaxial bending. Fig. 7(d) shows the OP loads ( and ) applied to the and plates, as well as the undeformed and deformed shapes.
Because the plate thickness is small relative to the perimeter dimensions, the OP behavior of IATPs is characterized using multiple beam elements (i.e., to capture biaxial bending) (Choi and Park 2010). Fig. 8 shows the beam idealization of an IATP element. The beam elements, which represent a strip of the plate, are along the directions parallel [fiber-parallel (FPa)] and perpendicular [fiber-perpendicular (FPe)] to the fibers. The contribution to OP bidirectional bending from each fiber-parallel and fiber-perpendicular beam is and , respectively and is reflected in the stiffness matrix of the IATP element. Assuming that the fiber-parallel and fiber-perpendicular directions have and strips, respectively, it follows that , where and . In Appendix V, the contribution factors and are analytically calculated for a single IATP element. However, for more complex assemblies with large numbers of DOFs, a numerical solution is required (Bozorgnia and Bertero 2004).
The OP behavior of each IATP is represented by that of the beam (strip) elements. To calculate the stiffness matrix of each beam (), the end nodes are indexed as 1 and 2 in Fig. 9(a) and the OP behavior is described using 6 DOFs in the GCS. The vector describes the kinematics associated with these DOFs when the element responds within the linear elastic range. The components of the vector are shown in Fig. 9(a). The terms , , , and are the translations, and and are the rotations at the end nodes. Similar to the vector, the element force components are described by the global vector [Fig. 9(b)]. The details of the formulation of the beam element used to characterize the OP behavior and the corresponding force-deformation relationship are provided in Appendix VI.
The stiffness matrix for each beam element, , iswhere = matrix representing the partial derivative of the OP displacements with respect to ; = modulus of elasticity; = moment of inertia; and = initial length of the element.
(5)
Once is determined, the principle of superposition is used to obtain the stiffness matrix for the entire IATP. This is taken as the sum of the stiffnesses of the IMAs and the strip elements along the fiber-parallel and fiber-perpendicular directions while accounting for the contribution of each strip element to the OP behaviorwhere and = stiffness matrices for the beams parallel and perpendicular to the fiber orientation, respectively; and = associated flexural stiffness contribution factors; and , , and capture the semirigid behavior of the TT joints under tensile and flatwise shear loads and out-of-plane moments, respectively.
(6a)
(6b)
According to Nilson et al. (2010) and Hillerborg (1996), computing the flexural stiffness contribution of the orthogonal strips analytically is approximate. In fact, the load path is determined by multiple parameters, including the boundary conditions, distribution of the surface and edge loads, and the material orientation. Therefore, numerical simulations, which are described in the verification section, are needed to calculate and , and, consequently, the OP stiffness matrix of the strips and their contribution to the overall OP behavior.
Macromodels for IATPs
This section introduces the geometric development and associated mechanical properties of the macromodel.
Geometric Development: From Fabrication Contours to Macromodel Element
Using the fabrication contours of an IATP [Fig. 10(a)], the midsurface of each 3D plate is established [Fig. 10(a)]. Next, the macromodel is generated from the midsurface using four main steps: (1) using the corner nodes of the midsurface to create a planar polygon (hereafter called boundary elements) [Fig. 10(b)], (2) identifying the location of the IMAs in the boundary elements [Fig. 10(c)], (3) dividing the polygon into equal segments along the fiber-parallel and fiber-perpendicular directions and adding inner lines at the interface between segments (hereafter called inner beam elements) [Fig. 10(d)], and (4) splitting the boundary elements at each intersection and connecting the end nodes of the inner beam elements, TT joints, and the contour corners [Fig. 10(e)].
Macromodel and Associated Mechanical Properties
The proposed macromodel [Fig. 11(c)] is formulated to be compatible with different quadrilateral geometries and load cases. It consists of boundary elements [Fig. 11(d)], inner beam elements [Figs. 11(e and h)], two (uniaxial) shear springs [Figs. 11(j and k)], two-node link elements representing the joints [Fig. 11(g)], and uniaxial springs distributed along the boundary elements [Figs. 11(f and i)], which capture the tension–compression behavior of the plate. Overall, structural stability is maintained by connecting the inner and perimeter beam elements, the uniaxial springs, and the IMAs.
The boundary elements are pin-ended and free to rotate about their local - and -axes [Fig. 11(d) for the LCS]. However, the fiber-parallel [Fig. 11(e)] and fiber-perpendicular [Fig. 11(h)] inner beams have a flexural release about their local and , respectively. In other words, the boundary elements [Fig. 11(d)] are free to rotate in the IP and OP directions, whereas the inner beams [Figs. 11(e and h)] provide OP flexural resistance and are only free to rotate in the IP direction. It is within this context that the macromodel incorporates the mechanical properties of the inner beam elements to simulate the OP kinematic behavior and is independent of the boundary elements. To simulate the IP kinematic response, Eq. (4) and two modeling strategies are used: (1) a pair of uniaxial springs in the fiber-parallel and fiber-perpendicular directions is used to simulate the shear behavior () [Figs. 11(j and k)]; and (2) a set of distributed uniaxial tension–compression springs along the edges of the quadrilateral macromodel [Figs. 11(f and i)] is used to simulate the combination of direct axial () and flexural () loads. These two modeling strategies were also adopted by Christovasilis and Filiatrault (2013), Casagrande et al. (2016), and Caliò et al. (2012).
In IATP structures, the joints and plates are one unit. In other words, because no additional connectors such as screws or fasteners are employed, the joints and the associated plate cannot be separated. To better understand their behavior, experiments were performed by Roche (2017) and Rezaei Rad et al. (2019a, b), which included both the joints and the neighboring region of the plates in the test specimen. With this type of experiment, the behavior of what is referred to as the joint region is examined and the initial stiffness from an idealized backbone curve is assigned to the two-node link element [Fig. 11(g)]. Consequently, the use of link elements to explicitly describe the behavior of the joint region can account for the local behavior of IATPs.
Because the IMA boundary element behavior is simulated using the aforementioned two-node link elements [Fig. 11(g)], the primary role of the boundary beams is to provide overall stability, and are therefore modeled as rigid. These rigid boundary elements are also pin-ended so that they can freely rotate about their local IP and OP axes [Fig. 11(d)]. Moreover, by using pin-ended rigid boundary elements, the actual geometry of the timber plate is represented in the macromodel. Additionally, because the end nodes have flexural releases [Figs. 11(e and h)], the behavior of the macromodel is independent of the cross-section size of the boundary elements. Therefore, the inner beam elements and uniaxial shear and tension–compression springs encapsulate the mechanical properties of the macromodel. This modeling strategy makes the macromodel compatible with different polygon geometries and load cases. A similar strategy was adopted by Caliò et al. (2012), Caliò and Pantò (2014), and Pantò et al. (2017).
To determine the cross-section properties of each inner beam element, the associated tributary area [Figs. 11(e, f, h, and i)] is computed, and consequently, the stiffnesses of the shear springs, tension–compression springs located around the edges, and the inner beams are determined. The IP flexural stiffness of the plate [ in Eq. (4b)] is implicitly represented using the aforementioned uniaxial tension–compression springs [Figs. 11(f and i)]. The flexural stiffness of the plate can be expressed by the uniaxial tension–compression springs (Fig. 12) (Caliò et al. 2012; Pantò et al. 2016). In other words, in addition to simulating the tension–compression behavior, these springs can capture the IP flexural deformation of the element (Fig. 12).
Macromodel Verification and Discussion of Results
A three-step procedure was used to verify the macromodel. The plate level behavior was first verified against the detailed FE models for IP and OP loading cases. Second, the macromodel was verified against the response from physical experiments on full-scale subassembly specimens performed by Nguyen et al. (2019a). The results from a detailed FE model were also included in this comparison.
Construction Material
Beech BauBuche LVL (Blaß and Streib 2017) (Pollmeier, Thuringia, Germany) was used as the construction material. Each timber board consisted of 13 3-mm-thick peeled veneer layers, 15% of which (2 of 13 layers) were aligned perpendicular to the fiber direction (referred to as cross-plies). Table 1 summarizes the mechanical properties of the BauBuche LVL material. LVL boards are engineered timber products with homogeneous and orthotropic material properties (Roche 2017; Stitic et al. 2019; Hassanieh et al. 2017).
Symbol | Description | Value |
---|---|---|
Mean modulus of elasticity, 0° | 1,3200.0 | |
Mean modulus of elasticity, 90° | 2,200.0 | |
Mean edgewise shear modulus, 0° | 820.0 | |
Mean edgewise shear modulus, 90° | 430.0 | |
Edgewise bending, 0° | 60.0 | |
Edgewise bending, 90° | 10.0 | |
Compression, 0° | 53.3 | |
Compression, 90° | 19.0 | |
Tension, 0° | 51.0 | |
Tension, 90° | 8.0 | |
Edgewise shear, 0° | 7.8 | |
Edgewise shear, 90° | 3.8 |
Description of FE and Macromodels and Analyses
The detailed FE model was constructed using Abaqus/CAE version 6.14. The midsurface of each timber panel was represented by a polysurface element which was divided into meshes and represented using shell elements. Only linear elastic response was considered for the shell element. Orthotropic material properties (Table 1) were assigned to the shell elements. A quad configuration and finite strain S4R elements were used for meshing. The mesh size was smaller near the connections, with a seed size of 20 mm, and larger in the other areas of the IATP, with a seed size of 50 mm. The connectors used to simulate the TT joints were modeled using the Bushing element.
The macromodels were constructed in OpenSees version 2.5.0 (Mazzoni et al. 2013). A uniaxial elastic material with infinite stiffness was assigned to the boundary elements. An elastic beam–column element was used for the inner beams based on their associated tributary area and the timber properties for each orthogonal direction. The dovetail joints, which form a continuous connection between plates and , were assumed to be rigid (Nguyen et al. 2019a). The connectors used to simulate the TT joints were modeled using a two-node link element. To simplify the modeling process, the same division used to define the inner beam elements was employed to determine the location of the uniaxial tension–compression springs, which were distributed around the perimeter. Consequently, these springs were placed at the end nodes of each inner beam element.
Model Verification at Plate Scale
Fig. 13(a) shows the geometric configuration of the IATP plates inside a box, which is based on the average dimensions of the elements in the full-scale prototype (e.g., Fig. 2). Because of the assembly process for a four-sided box, the and plates are subjected primarily to IP loads, whereas the and plates are subjected to OP loads. Therefore, the macromodels representing the IP behavior of the and plates [Fig. 13(b)] and the OP behavior of and plates [Fig. 13(c)] were constructed and the load–deformation behavior was compared with that of the corresponding detailed FE models. For the macromodels, Fig. 14 shows the section size of the inner beam elements, as well as the stiffness values for the uniaxial tension–compression and shear springs. The inner beam elements were generated by dividing the and plates into three equal segments along the fiber-parallel and fiber-perpendicular directions. Furthermore, the and plates were divided into three and five equal segments along the fiber-parallel and fiber-perpendicular directions, respectively. The sensitivity of the load–deformation behavior to variations in the number and configuration of inner beam elements (Fig. 15) was evaluated, and the results are discussed in the next two subsections.
Results for IP Load Case
For the IP load case [Fig. 13(b)], the force corresponding to the ultimate limit state (ULS) is calculated using Eq. (7) (CEN 2008). The shear strength of the timber boards is the weakest resisting element and control the behavior (Table 1)where and = applied stress and design strength values, respectively = effective cross-section area; = shear force; = characteristic shear strength (Table 1); and , , and = modification factors (Table 2) in accordance with CEN (2008).
(7)
Symbol | Description | Value |
---|---|---|
Crack modification factor | 1.0 | |
Material modification factor | 0.8 | |
Strength modification factor | ||
Partial factor | 1.3 |
Source: Data from CEN (2008).
The IP ULS load is then computed and distributed along the element height at each IMA [Fig. 13(b)]. The boundary conditions and location of the applied loads are shown in Fig. 13(b).
To compare the response of the FE and macromodels, the deformed shape along the element height for the different configurations defined in the sensitivity analysis are presented in Fig. 16. The deformed shape obtained from the macromodel was similar to the one obtained from the FE model. However, the deflections in the macromodel were slightly higher in the lower portion of the element and slightly lower in the upper portion. These differences are attributed to the different element formulations in the macro and FE models. Specifically, the FE model takes advantage of plate theory, in which nonlinear partial differential equations with an order of 8 govern the plate kinematics (Reddy 2007). In the element formulation, the IP shear, IP bending, and axial deformation are coupled. This leads to differences in the curvature along the element for the FE and macromodels, in which the latter used separate (uncoupled) springs to model the shear, flexural, and axial behavior of the plates [off-diagonal components of matrix in Eq. (4)]. Furthermore, in the FE model, the rigidity of the element is distributed over the plate, whereas in the macromodel, the rigidity is concentrated in a specific number of discrete elements, i.e., the shear and tension–compression springs. Nonetheless, the average difference between the horizontal displacement in the FE and macromodels was 9.01%, which is an acceptable degree of accuracy. The horizontal displacement values for both the macro and FE models in the joint regions identified in Fig. 13(b) are provided in Table 3.
Results for OP Load Case
For the OP load case, both the TT joints and timber plates play a role in the load-transfer mechanism. Generally, joints are the weakest elements in IATP structures (Rezaei Rad et al. 2019a). Specifically, the ultimate strength of IMAs is lower than that of timber plates (Rezaei Rad et al. 2019b). Therefore, the IMAs control the ULS of the OP load case [Fig. 13(c)]. Both the macro- and FE models were subjected to the OP load corresponding to the ULS of the TT joints (Rezaei Rad et al. 2019a, b; Nguyen et al. 2019b). The OP loads applied at the TT joints are shown in Fig. 13(c). The mechanical properties of the two-node link elements used to simulate the TT joints are presented in Table 4.
Symbol | Description | Value |
---|---|---|
() | Tensile stiffness | 416.81 |
() | Compressive stiffness | |
() | IP shear stiffness | 15,009.24 |
() | OP shear stiffness | 9,489.04 |
() | Torsion stiffness | |
() | OP flexural stiffness | |
() | IP flexural stiffness | 170,000.0 |
The undeformed and deformed shapes of the macromodels, including the configurations defined in the sensitivity analysis, and the FE model are shown in Fig. 17. Similar to the IP load case, the behavior captured by the macromodel was similar to that of the FE model. The average difference between the displacement in the FE and macromodels was 8.3%. The displacement values for both the macro and FE models in six different joint regions [Figs. 13(c) and 17] are provided in Table 5.
Assembly-Level Model Verification
The macromodel was verified against the results of a three-point bending test performed on a IATP subassembly, which was carried out at IBOIS. Three replicates of the prototype were tested by Nguyen et al. (2019a). The full-scale prototype, which represented a portion of an IATP structure, included 15 boxes. LVDTs were installed at the midspan of the assembly to record the displacements. The structural dimension, test setup, and the load–deformation curves are shown in Fig. 18.
Portions of the macro and FE models are shown in Fig. 19. The mesh size and element type noted in the description of the FE and macromodel section was adopted for the subassembly verification. For the macromodels, the section size of the inner beam elements and the stiffness values for the uniaxial tension–compression and shear springs are shown in Fig. 14.
The macro and FE models were subjected to the same loading protocol used in the physical test. Using the deflection-based criteria (span length/300), the responses were assessed at the serviceability limit state (SLS) to ensure a linear elastic response. Two LVDTs were installed on the north and south sides of the physical specimen (Fig. 18), and a force transducer was attached to the load actuator (hydraulic jack). Accordingly, the load–deformation curves were obtained from the macro and FE models and the physical experiments (including the average response from the three replicates). Figs. 20(a and b) show the load–displacement curves associated with the LVDTs installed on the north and south parts of the physical specimen. The results demonstrate that the responses obtained from both the macro and FE models were very similar to the experimental load–deflection behavior. The average linear stiffnesses computed from the load–deflection curve in Fig. 20 for the macro () and the FE () models were similar in value to that of the physical specimen (). The stiffness values of the macro and FE models were and different, respectively, from the experimental stiffness value.
Evaluating Relative Contribution of Different Element Groups to Overall Response
Considering the joints, plate types (, , , and ), and the fiber orientation (fiber-parallel and fiber-perpendicular), 13 different element groups existed in the model domain (6 for the inner beams, 6 for the uniaxial springs, and 1 for the joints). To understand the contribution of the inner beams to the overall response, one element from each group and the associated local coordinate system is shown in Fig. 21(a). The local forces for each element group, including axial and shear and torsion and bending moments, were first calculated. Given the section properties (area and moment of inertia), the associated stresses were computed. Fig. 21(b) compares the absolute maximum stress for each element group. The overall response was governed primarily by the IMAs, and the role of the timber plates was less important (Fig. 21). Furthermore, among the DOFs associated with the response of the IMAs, the IP behavior (DOFs 1 and 2 in Fig. 21) governed the response of the full-scale prototype. The stress associated with these DOFs was 4.4 and 7.49 MPa, respectively. This state of stress is consistent with what was reported by Rezaei Rad et al. (2019a, b), and Nguyen et al. (2019b) for the SLS. In addition, the top and bottom timber plates ( and ) had a greater influence on the response compared to the cross plates ( and ). This was directly related to the assembly process used for IATPs, in which each box is connected to its neighbor via the TT joints of the top and bottom plates. For DOF 6, which corresponds to the OP bending of the inner beams, the and plates had the greatest influence on the deformation, with stress values of 1.36 and 2.88 MPa for the fiber-parallel and fiber-perpendicular directions, respectively. In other words, the top and bottom plates were primarily responsible for resisting the OP loads. Regarding the cross plates, the plate was also subjected to the OP loads, with a stress of 1.65 MPa [DOF 6 of the fiber-perpendicular inner beam, CL-FPe, in Fig. 21(b)]. This is attributed to the fact that the four-sided box is geometrically asymmetric, and therefore, the cross plates also carry OP loads.
Computation Duration for Macro and FE Models
In addition to comparing the load-displacement response of the macro and FE models with that of the experiment, the time required to construct the model (data creation runtime) and run the analysis (analysis runtime) was evaluated. As stated previously, the TT joints govern the overall response of the system, and therefore, the FE model increases the computational complexity without improving the efficiency. In addition to the analysis runtime, the creation of the model database is very time consuming for the FE models. For the mesh size and configuration used in the verification studies, the model creation and analysis runtimes were 4 min 10 s and 8 min 36 s, respectively using an Intel Core i7-4800MQ CPU @ 2.7GHz with 16 GB of RAM. In contrast, the macromodel was constructed and analyzed in only 34 s.
Conclusions
A macromodeling technique for spatial timber plate structures with integral mechanical attachments (IMAs) was presented. The joinery process was described, and the kinematics and load-transfer mechanism for the integrally attached timber plate (IATP) elements were examined. The overall behavior of an IATP component was deemed to be governed by the in-plane and out-of-plane plate response. Consequently, the kinematic degrees of freedom and the corresponding force vectors were defined. For the IP response, the displacement field includes contributions from axial, shear, and flexural deformation and rigid-body translation and rotation. For the OP response, displacements are associated with flexural deformations and rigid-body translation. Applying the virtual work principle, the IP and OP element stiffnesses were formulated. It was shown that (1) the IP stiffness of a timber plate equipped with TT IMAs can be simulated by considering the axial, shear, and flexural stiffnesses of the timber plate and the semirigid behavior of the TT joints, and (2) the OP behavior of IATPs can be captured using multiple beam elements, and the stiffness of the entire plate is computed using the principle of superposition.
Using the fabrication contours of an IATP element, the geometry of the macromodel was generated. The model consists of inner beam elements, boundary elements, two (uniaxial) shear springs, two-node link elements representing the joints, and uniaxial springs distributed along the boundary elements, which represent the tension–compression behavior of the plate. The mechanical properties associated with each component were identified.
The behavior of the macromodel was verified against detailed finite-element models at the plate level for both the IP and OP load cases. The deformed shape obtained from the macromodel was similar to that of the FE model. The macromodel was then verified against the response of a three-point bending experiment performed on an IATP subassembly and the corresponding FE model. The results showed that the responses obtained from both the macro- and FE models were similar to those of the experiment. Using only beams and spring elements, the macromodel analysis converged with no instabilities, and the associated computational time was only 34 s. However, depending on the element type, the adopted integration scheme, and the mesh size, the FE models were susceptible to variations in the response and numerical instabilities. This is because the one-dimensional (1D) wire elements used to simulate the TT joints were connected to 2D shell elements with high mesh density. Therefore, as also shown by Caliò et al. (2012) and Pantò et al. (2016), the detailed FE simulations do not necessarily provide improvements in terms of accuracy. Furthermore, the total computational time for the FE models, including the creation of the model database and runtime, was 12 min 46 s, which was significantly more than that of the macromodel. It was determined that the joints contributed the most to the overall behavior of the system. Furthermore, the OP behavior of the timber plates had the second highest contribution to the response of the IATP structure.
There are some limitations of the proposed macromodeling approach. As with the FE method, compatibility is maintained at the nodes. However, because the beam ends are located at the nodes located along the perimeter of the plates, the modeling approach does not permit the determination of the deflection and force values on the interior of the plates. This can be improved by using multiple segmented elements for each beam. Furthermore, detailed FE models provide a reliable evaluation of the local stress–strain response especially at small scales (e.g., at the connections), which is not necessarily reflected in the macroscopic modeling approach.
This study formulated a macromodel for linear elastic analysis. However, previous experiments showed that material nonlinearity can occur in the joint regions. Therefore, as a possible extension, springs with idealized multilinear backbone curves can be introduced to the macromodel to account for material nonlinearity and to capture the postpeak-force behavior. Additionally, the element stiffness matrix and equilibrium equations were formulated considering the undeformed configuration. However, in large-span IATP structures, the effect of axial loads would be more pronounced, and, accordingly, geometric nonlinearity could play a significant role. Therefore, another possible extension could be to formulate the macromodel to incorporate geometric stiffness, whereby equilibrium is satisfied on the deformed configuration. Specifically, the general corotational formulation can be used to establish the compatibility and equilibrium relationships in the deformed configuration.
Appendix I. Mathematical Formulation of Global Vector of Single IATP Element
Assuming consecutive deformations and displacements, including rigid-body translations, shear deformations, flexural deformations, and rigid-body rotations, Fig. 22 shows the vector relationship between and . The coordinate vectors, corresponding to the recognized displacement and deformation fields, are those associated with shear deformation , flexural deformation , and rigid-body rotation (Fig. 22). As noted previously, the LCS of an IATP element changes at each step, and the coordinate vector should be updated. The rigid-body translations do not cause any translation or rotation in the LCS, and therefore the coordinate vectors remain unchanged after they are applied. Recalling Eq. (1), the displacement vector is re-expressed in terms of the recognized IP DOFs and the local coordinate vectors aswhere describes rigid-body translation (); is associated with rigid-body rotation (); is related to the uniform flexural deformation (); and describes shear deformation (). Assuming a uniform IP flexural deformation, it follows that . Therefore, Eq. (8) is reformulated aswhere = transformation matrix for rigid-body rotation. Referring to Fig. 22, , and therefore, Eq. (9) is rewrittenwhere = transformation matrix for IP flexural deformations. Referring to Fig. 22, it follows that , and therefore, Eq. (10) is rewritten
(8)
(9a)
(9b)
(10)
(11)
Given that and , Eq. (11) is written
(12)
Applying and
(13a)
(13b)
Finally, the total deformation of an IATP element in a GCS can be expressed in terms of LCS indicators as
(14)
Appendix II. Mathematical Formulation of Partial Derivative System of Equations of Matrix
Matrix is formulated aswhere
(15)
(16a)
(16b)
(16c)
(16d)
(16e)
(16f)
(16g)
Appendix III. Virtual Work Principles for IP Formulation of IATP Elements
According to the virtual work principle, it is assumed that the external work is equal to the internal work
(17)
The external work is done by the external force, , on the virtual displacement, , whereas the internal work is done by the compatible virtual deformations/strains ( and for timber plates, and for IMAs) on the element forces/stresses (the axial stress and shear stress for the timber plate, and for the IMAs); and are shear and axial strains at Point M in Fig. 23. Accordingly, and are the shear and axial stresses. Eq. (17) then is rewrittenwhere determines the IMA contributions; determines the shear contribution of the plate; and determines the axial-flexural contribution of the plate to the IP response.
(18a)
(18b)
According to Fig. 23, the strains and stresses are function of the position along the element axis and the position of Point M within the cross section (Bozorgnia and Bertero 2004). Regarding the shear contribution, the shear strain at Point M in Fig. 23, , is written as a function of section deformation and strain distribution
(19)
A uniform distribution of strain and deformations along the profile is assumed (Christovasilis and Filiatrault 2013), which results in , where is defined in Fig. 6.
Regarding the axial-flexural contribution, the axial strain at Point M in Fig. 23, , is written as a function of section deformation and strain distribution
(20)
Because Bernoulli’s assumption is valid for the IP flexural deformations, the axial strain distribution function results in (Bozorgnia and Bertero 2004). For the IP behavior, the rotation about the local -axis is ignored, and therefore . Therefore, Eq. (18b) is rewrittenwhere and = section forces for shear and axial virtual works, respectively. Because the stress is uniformly distributed, the section forces are and , where , , and are the shear and axial forces and bending moment about local -axis in Section A-A at location (Fig. 23). Eliminating the virtual terms, the applied loads are expressed in terms of internal forces in the matrix form
(21)
(22)
The same formulation can be developed for the transverse direction (local -axis).
Appendix IV. Mathematical Formulation of IP Stiffness of IATP Element
The stiffness of the plate element is defined as the ratio
(23)
Expanding Eq. (23), it follows that
(24a)
(24b)
The higher-order effects, which are reflected in , are neglected (Christovasilis and Filiatrault 2013), and the IP stiffness of an IATP elements is writtenwhere = axial stiffness along fiber-parallel direction; = IP shear stiffness associated with relative displacement along local -axis, and = flexural stiffness about the local - or local -axis. These stiffnesses are calculated in Fig. 11. The same equilibrium equations can be developed for the IP behavior of an IATP element along the fiber-perpendicular direction.
(25a)
(25b)
Appendix V. Role of Timber Orthotropic Material in OP Load-Carrying Mechanism of IATP Element
The contribution from loading in both directions is analytically addressed for a single and plate because it is subjected to bidirectional bending [Fig. 7(b)]. The most convenient way to do so is to select two orthogonal strips which share a common point, i.e., they intersect. In Fig. 24, and , which belong to the longitudinal and transverse directions, respectively, are selected. It is assumed that the surface and edgewise loads, denoted , are uniformly applied to a and plate. Portions of this uniformly distributed load, which are applied to and , are denoted and , respectively. Because the corner point (the reference point in Fig. 24) is a part of the same monolithic plate and is affected by both and , its deflection should be the same in the two orthogonal strips. Using beam theory and assuming that the strips are fixed at their supports (Fig. 24), the corner deflection associated with the two orthogonal strips is computed using a fixed-end cantilever beam assumption [Eq. (26)]. This concept is adapted from the strip method used to design reinforced concrete structures (Hillerborg 1996; Nilson et al. 2010)where , , , , , and are defined in Fig. 24; and and = contributions to bending from and , respectively. Assuming that the longitudinal and transverse directions have and strips, respectively, each with the same width, it follows that . Accordingly, Eq. (26) can be rewritten
(26)
(27)
The plate dimensions are designed during the architectural CAD model development. The length and width of the plates typically are very similar (Robeller et al. 2016). Because the modulus of elasticity in the fiber-parallel direction ()generally is 5–6 times that in the fiber-perpendicular direction () (Blaß and Streib 2017; Finland VTT Expert Services 2016), the implementation of Eq. (27) is such that 84%–86% of the surface loads are resisted by the former.
Appendix VI. OP Formulation of Strip Elements
It was shown that the OP kinematic of an IATP element can be represented by multiple strip (beam) elements. The OP behavior of a strip element now is formulated using the beam theory. The formulation is presented for the OP bending about the local -axes (Fig. 25); the same methodology can be used to describe the OP bending about the local -axis. The undeformed strip element has the initial length of . The rigid-body rotation of the element and the deformed length are denoted and , respectively.
The OP displacement field of a strip element about the local -axis is described by vector in the basic reference system [Fig. 26(a)], where is the axial elongation, and and are the nodal rotations. These components are associated with the basic forces of a beam element, [Fig. 26(b)]. The relation between and is described by the basic stiffness matrix, in Eq. (28) (Bozorgnia and Bertero 2004)
(28)
Applying the virtual work principle, the element force vector is expressed by the internal force vector in
(29a)
(29b)
Where the vector is defined as the virtual displacement vector. The matrix describes the partial derivative system of equations [Eq. (30)].
(30a)
(30b)
The stiffness matrix of the strip element then is calculated as
(31)
Neglecting the higher effects [], the stiffness matrix of a strip element can be expressed as
(32)
Data Availability Statement
Some data, models, or code generated or used during the study are available from the corresponding author by request.
Acknowledgments
The support of the National Centre of Competence in Research (NCCR) Digital Fabrication funded by the Swiss National Science Foundation (SNSF) is acknowledged (Agreement ID #51NF40141853). The support of École Polytechnique Fédérale de Lausanne (EPFL) also is acknowledged with many thanks.
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Received: Aug 27, 2019
Accepted: Feb 28, 2020
Published online: Jul 20, 2020
Published in print: Oct 1, 2020
Discussion open until: Dec 20, 2020
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