Timber Plates Research Articles

Polynomial chaos based rational approximation in linear structural dynamics with parameter uncertainties

Surrogate models enable efficient propagation of uncertainties in computationally demanding models of physical systems. We employ surrogate models that draw upon polynomial bases to model the stochastic response of structural dynamics systems. In linear structural dynamics problems, the system response can be described by the frequency response function. It is well known that standard polynomial chaos expansions of the frequency response present slow convergence around system eigenfrequencies, due to the highly nonlinear nature of the frequency response for low damping. To overcome this issue, we develop a rational approximation that expresses the system response as a rational of two polynomials with complex coefficients. To estimate the latter, we propose a regression approach that is non-intrusive and can be easily coupled with existing deterministic solvers. We demonstrate the effectiveness of the proposed method with two examples, a two-degree-of-freedom system and a finite element model of a cross laminated timber plate.

Predicting failure of notched cross-laminated timber plates including the effect of environmental stresses

ABSTRACT This paper derives analytical predictions for notched cross-laminated timber (CLT) plate delamination using fracture mechanics for crack propagation from the notch root. The analysis accounts for both CLT heterogeneity and residual stresses caused by seasonal variations in temperature and humidity. The accuracy of predictions coupled with shear corrections, was confirmed by comparison to finite element analysis. The fracture mechanics approach was validated as appropriate by comparison to recent experiments on notched CLT beams. Residual stress effects are significant and depend strongly on notch depth. Environmental changes that promote swelling, such as increases in temperature or moisture content, are more detrimental to plate integrity than are changes that promote shrinkage. Relatively modest levels of swelling stress can potentially cause spontaneous delamination. A new design guide for notched CLT plates that includes effects of heterogeneity and residual stresses is proposed.

Displacement Study of a Large-Scale Freeform Timber Plate Structure Using a Total Station and a Terrestrial Laser Scanner.

Recent advances in timber construction have led to the realization of complex timber plate structures assembled with wood-wood connections. Although advanced numerical modelling tools have been developed to perform their structural analysis, limited experimental tests have been carried out on large-scale structures. However, experimental investigations remain necessary to better understand their mechanical behaviour and assess the numerical models developed. In this paper, static loading tests performed on timber plate shells of about 25 m span are reported. Displacements were measured at 16 target positions on the structure using a total station and on its entire bottom surface using a terrestrial laser scanner. Both methods were compared to each other and to a finite element model in which the semi-rigidity of the connections was represented by springs. Total station measurements provided more consistent results than point clouds, which nonetheless allowed the visualization of displacement fields. Results predicted by the model were found to be in good agreement with the measurements compared to a rigid model. The semi-rigid behaviour of the connections was therefore proven to be crucial to precisely predict the behaviour of the structure. Furthermore, large variations were observed between as-built and designed geometries due to the accumulation of fabrication and construction tolerances.

Frequencies Prediction of Laminated Timber Plates Using ANN Approach

Frequencies Prediction of Laminated Timber Plates Using ANN Approach

On the diffuseness of the vibrational field of a cross-laminated timber plate: Comparison between theoretical and experimental methods

The modelling of vibro-acoustic properties of plates usually requires the hypothesis of diffuse conditions of the vibrational field. For heavy-weight homogeneous isotropic plates, this condition is met starting from low frequencies. Nevertheless, the same hypotheses do not hold when applied to stiff, inhomogeneous and non-isotropic materials, like wood, thus its modal behavior needs to be addressed with more detail. The aim of the work is to benchmark theoretical methods and experimental measurements that allow to estimate when the transition from modal to diffuse field occurs in Cross Laminated Timber (CLT) elements. For this purpose, the diffuseness of the vibrational field has been investigated through the experimental evaluation of the mode count and the modal overlap factor on a pilot CLT panel. Measurements of the spatial distribution of modes, impulse responses, dispersion relations and driving point admittance were conducted in laboratory on an 8 m2 CLT plate. The analysis of the results allowed, on the one hand, to directly estimate the two parameters, and on the other, to provide input data necessary to evaluate and compare different theoretical models. Results of direct measurements, performed up to 550 Hz, show that the number of modes occurring in one-third octave bands exceeds the standard threshold of 5 only at 500 Hz, whereas the modal overlap factor is lower than the standard threshold of 1 for most of the modes. This indicates that the vibrational field is mainly non-diffuse in the mid-low frequency range, due to the small number of modes combined with a low damping of the CLT plate. Nonetheless, measurements of modal reverberation time and structural reverberation time in one-third octave band provided consistent results, implying that modal decays are the major contributors to the one-third octave band decay. Furthermore, comparisons with theoretical models show that the driving point admittance proved to be the best method of analysis, requiring simple measurement setup and signal processing, whereas the ISO 10848-4 is affected by the thin plate hypothesis and underestimates the mode count at mid-high frequencies.

Experimental investigation on load bearing capacity of full scaled light timber framed wall for mid-rise buildings

Timber is an isotropic natural construction material with ductile and brittle behaviour in compression and tension, respectively. It is one of the most economical solutions for low-rise residential buildings, in many countries including Australia. Yet, its relatively lower load bearing capacity in compression perpendicular to the grain, and some consequences such as long-term and short-term shortening have resulted in limited applications of light timber framed system in multi-story buildings. In order to take the most advantage of this efficient and light-weight building material specifically light, applying some timber framed reinforcement solutions and better use of materials can improve the performance of timber framed structures. This study presents the results of a series of full scale experimental investigations on light timber framed walls under compression, whose results can be used for the design of mid-rise light timber framed buildings. First, with regard to the compression perpendicular to the grain, full scale walls connected through a stud-to-plate system and made of four different timber plate species (machine graded pine (MGP10), pine cypress, ash silver top, and cross laminated timber (CLT)) are tested. Then the effects of a reinforcement solution on enhancing the load bearing capacity is investigated. At the next stage, for studying the compression parallel to the grain, the performance of common stud-to-stud connections as well as a notched connecting system with machine graded pine (MGP10) and laminated veneer lumber (LVL) studs is considered. A comparison between the load bearing capacity of the tested systems is made, and the results are used in a case study on the design of a multi-story timber framed building. The results indicate that the solutions suggested for improving the compression bearing capacity can be applied to mid-rise light timber frame buildings.

Design framework for the structural analysis of free-form timber plate structures using wood-wood connections

Abstract Geometrically complex timber plate structures using wood-wood connections are increasingly designed thanks to digital fabrication. Whereas their fabrication has been automated using computer-aided design programming, research focused on the development of automated numerical tools for their structural analysis has been very limited. This paper presents a method to automate the generation of a finite element model for the analysis of structures with a large number of joints. The model is developed taking the example of recently developed double-layered timber plate shells. Vertical displacements obtained with the semi-rigid spring model proposed are in good agreement with experimental tests performed on small-scale prototypes, whereas rigid modelling of the connections highly underestimates displacements of the structure. Semi-rigidity of the connections in translation as well as in rotation was shown to highly influence the model. The generation of the numerical model was successfully automated for large-scale structures.

Sound radiation of cross-laminated timber panels based on a higher order plate theory

In this contribution, the problem of sound radiation by a rectangular baffled cross-laminated timber (CLT) plate excited by a point force is addressed. Due to the inherent properties of CLT, a higher order shear deformable plate theory is needed to capture its deformation field. To this end, a recently proposed equivalent single layer plate theory is extended to incorporate dynamic effects. The derived plate model is applied to flexural wave propagation in infinitely extended plates, revealing coincidence frequency bands of the considered CLT elements. For a simply supported rectangular plate, natural frequencies and total acoustic power output to the far-field are computed. Throughout all simulations, results obtained by the proposed higher order plate theory are set in contrast with those applying first order shear deformation theory and classical laminated plate theory, because those models are typically used for vibro-acoustic simulations of cross-laminated timber panels. Comparison of the natural frequencies shows that these quantities of the higher order theory are in excellent agreement with the outcomes of finite element simulations based on continuum elements. In contrast, the outcomes of the lower order models deviate with increasing frequency. Consequently, both lower order models fail at assessing acoustic power output in the low to mid frequency range when comparing results with the proposed plate theory.

Feasibility of predictive assessment of bending performance of CLT plates of Canadian hemlock

The correlation between the bending elastic modulus of lumbers along the primary direction and that of the resultant cross-laminated timber (CLT) plates in the full size suitable for slabs or wallboards was investigated to verify the feasibility of predicting the bending performance during the manufacturing of heavy building structures of this new type of material. A batch of Canada hemlocks lumber was graded based on a vibrational test that measures longitudinal elastic modulus. The elastic modulus and shear modulus in the transverse direction were also measured using the scheme of a torsional modal analysis of a cantilever plate. CLT were fabricated using the graded lumbers in sizes suitable for slabs or wallboards. The elastic moduli of these CLT products were measured using a conventional four-point static bending test. Finally, the static measurements of the elastic moduli of the CLT were compared with their predicted values that were calculated with the aforementioned data collected from the lumber pieces. The predicted elastic modulus along the primary direction of a CLT product agreed with the measured values. Therefore, the mathematical model of the CLT plate and the equation of its elastic modulus are feasible for the bending performance prediction in industrial production of CLT.

Experimental push-out investigation on the in-plane force-deformation behavior of integrally-attached timber Through-Tenon joints

Abstract Spurred by developments in digital fabrication, there has been an increase in the application of the Integral Mechanical Attachment (IMA) technique in timber plate structures. Despite its increase in popularity, there have been very few systematic investigations of the mechanical characteristics of IMA timber-to-timber connections. Focusing on in-plane shear, the force-deformation response of Through-Tenon joints is experimentally investigated, and their overall performance is assessed. The effect of influential design parameters is studied including the material type (Kerto-Q, and Beech-Q), fiber orientation (parallel and perpendicular to the loading direction), and the tab insertion angle (45°, 60°, and 90°). Twelve specimen groups, each with five replicates are tested. For the specimens with fibers oriented parallel to the loading direction (fiber-parallel specimens), a brittle failure is observed due to early rupture in the cross layers. On the other hand, more ductile behavior is observed for the specimens with fibers oriented perpendicular to the loading direction (fiber-perpendicular specimens) due to the compression forces perpendicular to the grains. Among the Kerto-Q specimens, the ones with a 60° tab insertion angle have the greatest in-plane strength. Moreover, an increase in the tab insertion angle results in a decrease in the yield deformation capacity and an increase in the slip modulus, ultimate deformation capacity, and ductility. Among Beech-Q specimens, the fiber-parallel cases and the fiber-perpendicular cases with a tab insertion angle of 60° and 90° have the highest in-plane strength, respectively. Also, the slip modulus increases with the tab insertion angle for the two fiber orientation cases. The ductility of the fiber-parallel Beech-Q specimens is unaffected by a change in the tab insertion angle. Overall, very small differences are observed between the yield and peak strengths and most specimens have been classified as having low ductility.

Modelling the viscoelastic mechanosorptive behaviour of Norway spruce under long-term compression perpendicular to the grain

Abstract The effects of variation in humidity coupled with long-term loading give rise to dimensional changes and creep effects in wooden elements. Many wooden products such as cross-laminated timber (CLT) plates as well as many common structural details used in timber engineering are vulnerable to variations in moisture content (MC) as well as to creep effects. This paper addresses the long-term effects in the material modelling of timber by the finite element method (FEM), also considering the viscoelastic and mechanosorptive effects in wood. The model was calibrated using both relaxation tests and creep tests. The results from both long-term compression perpendicular- to-grain tests (relaxation and creep) performed on glulam (GL30c) from Norway spruce (Picea abies) with moisture control are presented in this paper. The material model considers the effect of loading and moisture changes. For realistic comparison, the pith location of each lamella was specified in the numerical analyses. Ultimately, a comparison between the numerical results and the experimental results has been provided, exhibiting an overall good estimation of timber response.

Bending strength predictions of cross-laminated timber plates subjected to concentrated loading using 3D finite-element-based limit analysis approaches

Cross-laminated timber (CLT) is an innovative wood product with increasing utilisations. It is well known that the orthotropic and inhomogeneous strength properties of wooden boards have a strong influence on the load bearing capacity of CLT plates, especially when the complex wood fibre distribution due to randomly occurring knots is considered. Thus, high safety factors are used in current standards and a generally accepted numerical tool for the strength prediction of CLT plates is still not available. In this paper, we combine recent advances in 3D numerical limit analysis and a knot reconstruction algorithm, where not only the bending strength of CLT plates under concentrated loading is estimated using the numerical approach, but also the scatter of strength properties resulting from the material’s inhomogeneities is investigated using a stochastic approach. For the latter, data collected during the grading process of wooden boards are condensed into so-called strength profiles for single wooden boards. The limit analysis approach then allows a time-efficient simulation of a large number of randomly assembled CLT plates. The comparison of the resulting strength predictions to experiments shows good agreement with respect to both the mean load bearing capacity and the statistical scatter of strength.

Modelling the radiation efficiency of orthotropic cross-laminated timber plates with simply-supported boundaries

In this paper two prediction models to evaluate the radiation efficiency of orthotropic plates, developed with different approaches, are presented. A sound radiation model, based on an analytical/modal approach, is developed for a thin orthotropic plate, with the principal directions aligned with the edges. The model allows to consider the contribution of each mode, either resonant or non-resonant, as well as the influence of fluid loading on the plate dynamic response and on sound radiation. Moreover, a statistical model to evaluate the average radiation efficiency, based on a non-modal approach, which only considers the contribution of resonant modes, is presented. These two models have been used in order to predict the radiation efficiency of orthotropic cross-laminated timber (CLT) plates. CLT is an engineered wood material constituted by an odd number of lumber beams glued together, which have become very popular in the last twenty years in the building construction market. Due to their layered structure, CLT plates might exhibit a highly orthotropic behaviour. Both prediction models are validated by comparing the simulated results with the experimental radiation efficiency, obtained by means of vibro-acoustic measurements on three CLT plates. Finally, the influence of fluid loading on sound power radiated by CLT plates is investigated.

Explicit Vibration Formulas Applicable to Timber Plates with Free and Simply Supported Edges

Advances in research and development have led to the application of high-strength timber plates to flooring-slab construction. It is envisaged that floors built from timber plates will have different vibration behavior from conventional joisted floors. In this paper, explicit formulas are developed based on a matrix formulation that links both the symmetric and antisymmetric vibration modes of orthotropic plates. The developed solution can be used for the dynamic analyses of orthotropic timber floor slabs. The solution presented in this paper overcomes limitations associated with previous solutions by eliminating the need for approximate series expansions for solving the characteristic equations. Its application is demonstrated by considering responses of plates with all or two simply supported edges. The new solutions agree closely with classical and Finite-Element Method (FEM) solutions, as well as with the experimental results for Cross-Laminated Timber (CLT) plates. The advantage of the new approach versus alternatives is a combination of accuracy and computational efficiency.

Failure analysis of ribbed cross-laminated timber plates

The process of material failure i.e. cracks development and their propagation in an experiment related to the bending collapse of cross laminated timber plate with ribs is described. Numerical simulation of such an experiment by the nonlinear finite element method is presented. The numerical model is based on Hashin failure criteria, initially developed for unidirectional composites, and on material softening concept applied by the smeared crack approach. It is shown that such a numerical model can be used for an estimation of the limit load and the limit displacement of a cross laminated timber ribbed plate.

Experimental investigation of the influence of integral mechanical attachments on structural behaviour of timber folded surface structures

Structural behaviour of timber folded surface systems greatly depends on the connections ability to transfer the occurring forces between the adjacent elements and finally to the supports. This paper focuses on multiple tab-and-slot joints (MTSJ), where digital prefabrication is used to integrate connectors through plate geometry. Multiple plates assembled within a large scale folded surface structure were tested to examine the influence of connection detail type on its global structural behaviour. For this purpose an innovative test setup was devised that approximates uniformly distributed surface load. The connection details used were chosen with respect to preliminary small scale bending tests. Three groups of distinct large scale structures were tested: (1) structures with miter joint detail and adhesive applied along the edges; (2) structures with open slot MTSJ; and (3) structures with closed slot MTSJ. Extensive investigation into the load bearing behaviour and failure propagation for each of the three different types of structures has been conducted. For analysing their feasibility, the tested structures were also reviewed in terms of fabrication time, assembly and on-site construction. The obtained results show that even though adhesively joined structures provide highest structural stiffness, they exhibit multiple disadvantages when considering building scale applications. Open slot MTSJ structures results indicate that these joints cannot provide sufficiently reliable structural behaviour. Structures with MTSJ closed slots show that their joint geometry greatly improves both the ultimate load-bearing capacity as well as stiffness. Furthermore, by transferring the edge occurring forces mainly in compression, they provide additional ductility to the global system. Within the scope of this paper, closed slot MTSJ proved to be a very efficient connection type which can constitute a robust folded structural system made as a multiple assembly of thin timber plates.

Probabilistic sensitivity analysis to understand the influence of micromechanical properties of wood on its macroscopic response

This paper investigates the influence of the uncertainty in different micromechanical properties on the variability of the macroscopic response of cross-laminated timber plates, by means of a probabilistic sensitivity analysis. Cross-laminated timber plates can be modelled using a multiscale finite element approach which although suitable, suffers from high computational cost. Investigating parametric importance can incur considerable time penalty since conventional sensitivity analysis relies on a large number of code evaluations to produce accurate results. In order to address this issue, we build a statistical approximation to the code output and use it to perform sensitivity analysis. We investigate the effect of a collection of parameters on the density and Young’s moduli of wood. Additionally, the influence on the response of cross-laminated timber plates subject to bending, in-plane shear and compression loads is investigated due to its relevance within the engineering community. The presented results provide a practical insight into the importance of each micromechanical parameter, which allows research effort to be focused on important wood properties.

Double-layered timber plate shell

This article presents a new lightweight construction system for doubly curved shells made from timber plates, inspired by the masonry vaults of Eladio Dieste. The system was developed for a specific case-study project, and general applications to other freeform surfaces are being discussed. The shells are built from two interconnected layers of structural wood veneer plates, using integral through-tenon joints for a fast, precise, and simple assembly. This allows for the construction of a series of differently shaped vaults without a costly mold or support structure. Instead, inclined joints cut with a 5-axis computer numerical controlled milling machine embed the correct location and angle between plates into the shape of the parts. This constrains the relative motions between joined parts to one assembly path. To take advantage of the benefits of such connectors, the constrained assembly paths must be considered in the fundamental design of the system, allowing for the insertion of each plate. This imposes additional constraints in the segmentation process of doubly curved shells. In order to meet the requirements and resolve the multi-constraint system, we use a global, nonlinear optimization approach. Developed as a close collaboration between architects, computer scientists and structural engineers, this article includes an experimental analysis of the influence of parametric modifications in the shape of connectors on their load-bearing performance.

Determination of the elastic and stiffness characteristics of cross-laminated timber plates from flexural wave velocity measurements

Cross-laminated timber (CLT) is an engineered wood with good structural properties and it is also economically competitive with the traditional building construction materials. However, due to its low volume density combined with its high stiffness, it does not provide sufficient sound insulation, thus it is necessary to develop specific acoustic treatments in order to increase the noise reduction performance. The material's mechanical properties are required as input data to perform the vibro-acoustic analyses necessary during the design process. In this paper the elastic constants of a CLT plate are derived by fitting the real component of the experimental flexural wave velocity with Mindlin's dispersion relation for thick plates, neglecting the influence of the plate's size and boundary conditions. Furthermore, its apparent elastic and stiffness properties are derived from the same set of experimental data, for the plate considered to be thin. Under this latter assumption the orthotropic behaviour of an equivalent thin CLT plate is described by using an elliptic model and verified with experimental results.

Hybrid cross-laminated timber plates with beech wood cross-layers

A hybrid, three-layered, softwood-hardwood cross-laminated timber build-up with outer layers of European spruce (Picea abies) and a center cross-layer of European beech (Fagus sylvatica) has been investigated with regard to out-of-plane bending. The determination of the rolling shear properties of the beech cross-layer performed by different test and measurement methods comprising bending and compression shear tests was of primary interest. The shear capacity of the composite is significantly influenced by the spruce longitudinal shear strength at the beech-spruce interface. The characteristic values of rolling shear modulus and strength of the beech cross-layer from the bending tests were Gr,mean=350N/mm2 and fv,r,05=2.6N/mm2, respectively. Direct strain gauge measurements and compression shear tests resulted in 10–20% higher values. The high rolling shear properties render the shear lag implications of the softwood CLTs to a negligible quantity. The hybrid build-up can be designed as a rigid composite with small error versus a more exact analysis. The novel investigations reveal the great potential of mixed softwood-hardwood CLT build-ups for structural elements in the building sector.