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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
21

Lateral-Torsional Buckling Instability Caused by Individuals Walking on Wood Composite I-Joists

Villasenor Aguilar, Jose Maria 14 January 2013 (has links)
Recent research has shown that a significant number of the falls from elevation occur when laborers are working on unfinished structures. Workers walking on wood I-joists on roofs and floors are prone to fall hazards. Wood I-joists have been replacing dimension lumber for many floor systems and a substantial number of roof systems in light-frame construction. Wood I-joists are designed to resist axial stresses on the flanges and shear stresses on the web while minimizing material used. However, wood I-joists have poor resistance to applied lateral and torsional loads and are susceptible to lateral-torsional buckling instability. Workers walking on unbraced or partially braced wood I-joists can induce axial and lateral forces as well as twist. Experimental testing demonstrated that workers cause lateral-torsional buckling instability in wood I-joists. However, no research was found related to the lateral-torsional buckling instability induced by individuals walking on the wood I-joists. Furthermore, no research was found considering the effects of the supported end conditions and partial bracing in the lateral-torsional buckling instability of wood I-joists. The goal of this research was to derive mathematical models to predict the dynamic lateral-torsional buckling instability of wood composite I-joists loaded by individuals walking considering different supported end conditions and bracing system configurations. The dynamic lateral-torsional buckling instability was analyzed by linearly combining the static lateral-torsional buckling instability with the lateral bending motion of the wood I-joists. Mathematical models were derived to calculate the static critical loads for the simply supported end condition and four wood I-joist hanger supported end conditions. Additionally, mathematical models were derived to calculate the dynamic maximum lateral displacements and positions of the individual walking on the wood I-joists for the same five different supported end conditions. Three different lean-on bracing systems were investigated, non-bracing, one-bracing, and two-bracing systems. Mathematical models were derived to calculate the amount of constraint due to the lean-on bracing system. The derived mathematical models were validated by comparison to data from testing for all supported end conditions and bracing systems The predicted critical loads using the static buckling theoretical models for the non-bracing system and the static buckling theoretical models combined with the bracing theoretical models for the simply and hanger supported end conditions agreed well with the critical loads obtained from testing for the two wood I-joist sizes investigated. The predicted maximum lateral displacements and individual positions using the bending motion theoretical models for the simply and hanger supported end conditions agreed well with the corresponding maximum lateral displacements and individual positions obtained from testing for both wood I-joist sizes. Results showed that; a) the supported end condition influenced the critical loads, maximum lateral displacements and individual positions, b) the bracing system increased the critical loads and reduced the maximum lateral displacements, c) the critical load increased as the load position displaced away from the wood I-joist mid-span, d) the critical load reduced as the initial lateral displacement of the wood I-joist increased and e) the wood I-joist mid-span was the critical point in the dynamic lateral-torsional buckling instability. / Ph. D.
22

Avaliação da rigidez à flexão de toras de madeira por meio de vibração transversal / Evaluation of bending stiffness of wood logs by means of transverse vibration

Carreira, Marcelo Rodrigo 12 March 2012 (has links)
Antes de utilizar as toras de madeira como elemento estrutural é necessário avaliar as propriedades mecânicas desse material tanto por inspeção visual quanto por ensaio mecânico. A técnica de vibração transversal tem se destacado entre os demais métodos de Avaliação Não-Destrutiva para madeira serrada por obter estimativas acuradas do módulo de elasticidade à flexão. Contudo, testes prévios com essa técnica evidenciaram dificuldades de empregá-la na avaliação a rigidez à flexão de toras. O objetivo deste trabalho foi propor um método de ensaio para estimar o módulo de elasticidade na flexão de toras de madeira por meio de vibração transversal. O método proposto foi testado e validado em uma amostra de 40 toras de Eucalyptus sp. Foram obtidas boas correlações entre o módulo de elasticidade estático e o módulo de elasticidade dinâmico obtido pelo ensaio de vibração transversal. / Before using the wood logs as a structural element is necessary to evaluate the mechanical properties of this material by visual inspection and mechanical testing. The transverse vibration technique has stood out among the other methods of Non-Destructive Evaluation for lumber to obtain accurate estimates of the bending modulus of elasticity. However, previous tests showed difficulties with this technique to use it to evaluate the bending stiffness of logs. The aim of this work is to propose a test method to estimate the bending modulus of elasticity of wood logs by means of transverse vibration. The proposed method was tested and validated on a sample of 40 logs of Eucalyptus sp. It was obtained good correlation between the static bending and dynamic modulus of elasticity obtained by transverse vibration test.
23

Experimental and numerical study of the bending behaviour of textile reinforcements and thermoplastic prepregs / Etude expérimentale et numérique du comportement en flexion des renforts textiles et préimprégnés thermoplastiques

Liang, Biao 11 July 2016 (has links)
Cette thèse est consacrée à l'étude du comportement en flexion des renforts textiles et préimprégnés thermoplastiques par des méthodes expérimentales et numériques. Pour préimprégnés thermoplastique, aux températures élevées, la résine est à l'état fondu, et un glissement entre les fibres est possible. En conséquence la rigidité de flexion n'est pas directement liée au module de rigidité de traction dans le plan comme c'est le cas pour les matériaux continus classiques. Par conséquent, il est nécessaire de mesurer sa valeur par l'expérience. Un procédé de test de la rigidité à la flexion a été proposée pour les préimprégnés thermoplastiques. Il est réalisé dans une enceinte thermique. Une caméra CCD a été utilisé pour acquérir le profil de la déformation de flexion à différentes températures élevées en particulier au voisinnage du point de fusion. Le moment de flexion et la courbure ont été calculés le long de la ligne médiane de la profil de la déformation. La pente de la courbe moment-courbure est la rigidité à la flexion. Avec cette méthode, des essais de flexion ont été effectués sur plusieurs préimprégnés thermoplastiques. Pour simuler la déformation de flexion de matériaux fibreux épais, un élément de coque spécifique a été développée. Cet élément est constitué de segments de fibres continues. La rigidité de traction et de flexion de la fibre ont été prise en compte de dans l'énergie de deformation de cet élément. La forme curviligne locale a été construit pour tout segment de fibre a partir des voisins. Il a été utilisé pour caractériser les déformations de traction et de flexion pour le segment de la fibre. Plusieurs tests de simulation de flexion ont été réalisées avec cet élément de coque spécifique et ont été comparés avec les résultats expérimentaux pour montrer l'efficacité de cet élément proposé. Les résultats montrent cet élément de coque spécifique a une bonne capacité à simuler la déformation en flexion des matériaux fibreux épais. / This thesis is devoted to study the bending behaviour of textile reinforcements and thermoplastic prepregs by the experimental and numerical methods. At the high temperature, since the resin is melted, fibers would have the slippage between them, resulting the bending stiffness of thermoplastic prepreg is not directly related to its in-plane tensile modulus as the conventional continuous materials. Consequently, it's necessary to measure its value by the experimental method. A bending stiffness test approach was proposed for thermoplastic prepreg at elevated temperature. It was operated in an environmental chamber and a CCD camera was used to acquire the bending deflection shape. Bending moment and curvature were calculated along the midline of bending deflection shape. The slope of moment-curvature curve is the bending stiffness. With this method, bending tests were conducted for several types of thermoplastic prepregs at a range of high temperatures. In order to simulate the bending deformation of thick fibrous materials, a specific shell element was developed. This element was made of continuous fiber segments. Both the tensile and bending stiffnesses of fibers were taken into account. Local curve was constructed for any fiber segment and its two neighbors, which was used to characterize the tensile and bending deformations of fiber segment. Several bending simulation tests were performed with this specific shell element and were compared with the experimental results to show its efficiency. The results show this specific shell element has good capability to simulate the bending deformation of thick fibrous materials.
24

Bundles of Semi-flexible Cytoskeletal Filaments

Strehle, Dan 30 June 2014 (has links) (PDF)
Schaut man durch ein Mikroskop auf eine biologische Zelle mit angefärbten Zytoskelett, so erblickt man lange, mehr oder minder gerade Objekte. Mit ziemlicher Sicherheit gehören diese zu einer von drei Arten von Zytoskelettfilamenten -- Aktin- oder Mikrofilamente, Intermediärfilamente und Mikrotubuli. Schon seit mehreren Jahrzehnten versucht man die mechanischen Eigenschaften lebender Zellen nicht nur zu beschreiben, sondern ihr Verhalten von zwei tieferen Ebenen ausgehend zu verstehen: Inwiefern beschreiben die Eigenschaften von Filamentnetzwerken und -gelen die Zellmechanik und, noch tiefgreifender, wie bestimmen eigentlich die einzelnen Filamente die Netzwerkmechanik. Das Verständnis der Mechanik homogener und isotroper, verhedderter als auch quervernetzter Gele ist dabei erstaunlich detailreich, ohne jedoch vollständig dem jüngeren Verständnis von Zellen als glassartige Systeme zu entsprechen. In den letzten Jahren sind daher anisotrope Strukturen mehr und mehr in den Fokus gerückt, die die Bandbreite möglichen mechanischen Verhaltens enorm bereichern. Die vorliegende Arbeit beschäftigt sich mit solch einem hochgradig anisotropen System -- nämlich Aktinbündeln -- unter drei Gesichtspunkten. Mit Hilfe von aktiven Biegedeformationen wird ein funktionales Modul, das eine differentielle Antwort auf verschiedenen Zeitskalen liefert, identifiziert. Es handelt sich um Aktinfilamente, die durch transiente Quervernetzer gebündelt werden. Während sich das System nach kurz anhaltenden Deformation völlig elastisch verhält, sorgt eine Restrukturierung der Quervernetzer während langanhaltender Deformationen für eine plastische Verformung des Bündels. In einem weiteren Aspekt widmet sich die Arbeit der frequenz- und längenabhängigen Biegesteifigkeit. Die Methode des Bündel-Wigglings, das Induzieren von \"Seilwellen\", wird dabei genutzt, um aus der Wellenform die Biegesteifigkeit zu berechnen. Bündel von Aktinbündeln zeigen dabei ein Verhalten, das vom klassischen Worm-like-chain-Modell abweicht und stattdessen durch das Worm-like-bundle-Modell beschrieben werden kann. Der letzte Aspekt dieser Arbeit untersucht den Musterbildungsprozess bei der Entstehung von Aktinbündeln. Gänzlich unerwartet entstehen quasi-isotrope Strukturen mit langreichweitiger Ordnung, wenn der Bündelungsprozess erst nach der Polymerisation von Filamenten frei von zusätzlichen mechanischen Einwirkungen einsetzt. Da dieser Zustand nicht von der klassischen Flüssigkristalltheorie vorhergesagt wird, soll eine Simulation eine Hypothese zum Entstehungsmechanismus testen. Die Annahme einer lateralen Kondensation von Filamenten zu Bündeln reicht demnach aus, um die beobachteten Strukturen zu erzeugen. Diese Arbeit leistet somit einen Beitrag zum Verständnis hochgradig anisotroper Strukturen und deren Überstrukturen, wie sie auch in lebendigen Zellen reichlich vorhanden sind. / Being the most basic unit of living organisms, the cell is a complex entity comprising thousands of different proteins. Yet only very few of which are considered to play a leading part in the cell’s mechanical integrity. The biopolymers actin, intermediate filaments and microtubules constitute the so-called cytoskeleton – a highly dynamic, constantly restructuring scaffold endowing the cell not only with integrity to sustain mechanical perturbations but also with the ability to rapidly reorganize or even drive directed motion. Actin has been regarded to be the protagonist and tremendous efforts have been made to understand passive actin networks using concepts from polymer rheology and statistical mechanics. In bottom-up approaches isotropic, homogeneous actin-gels are well-characterized with rheological methods that measure elastic and viscous properties on different time scales. Cells, however, are not exclusively isotropic networks of any of the mentioned filaments. Rather, actin alone can already be organized into heterogeneous and highly anisotropic structures like bundles. These heterogeneous structures have only come into focus recently with theoretical work addressing bundle networks. and, in the case of the worm-like bundle theory, individual bundles. This work aims at characterizing bundles and bundle-crosslinker systems mechanically in two complementary approaches – in the time as well as in the frequency domain. In addition, it illuminates a bundle formation mechanism that leads to bundle networks displaying higher ordering.
25

Avaliação da rigidez à flexão de toras de madeira por meio de vibração transversal / Evaluation of bending stiffness of wood logs by means of transverse vibration

Marcelo Rodrigo Carreira 12 March 2012 (has links)
Antes de utilizar as toras de madeira como elemento estrutural é necessário avaliar as propriedades mecânicas desse material tanto por inspeção visual quanto por ensaio mecânico. A técnica de vibração transversal tem se destacado entre os demais métodos de Avaliação Não-Destrutiva para madeira serrada por obter estimativas acuradas do módulo de elasticidade à flexão. Contudo, testes prévios com essa técnica evidenciaram dificuldades de empregá-la na avaliação a rigidez à flexão de toras. O objetivo deste trabalho foi propor um método de ensaio para estimar o módulo de elasticidade na flexão de toras de madeira por meio de vibração transversal. O método proposto foi testado e validado em uma amostra de 40 toras de Eucalyptus sp. Foram obtidas boas correlações entre o módulo de elasticidade estático e o módulo de elasticidade dinâmico obtido pelo ensaio de vibração transversal. / Before using the wood logs as a structural element is necessary to evaluate the mechanical properties of this material by visual inspection and mechanical testing. The transverse vibration technique has stood out among the other methods of Non-Destructive Evaluation for lumber to obtain accurate estimates of the bending modulus of elasticity. However, previous tests showed difficulties with this technique to use it to evaluate the bending stiffness of logs. The aim of this work is to propose a test method to estimate the bending modulus of elasticity of wood logs by means of transverse vibration. The proposed method was tested and validated on a sample of 40 logs of Eucalyptus sp. It was obtained good correlation between the static bending and dynamic modulus of elasticity obtained by transverse vibration test.
26

Constitutive Modelling of Composites with Elastomer Matrix and Fibres with Significant Bending Stiffness / Constitutive Modelling of Composites with Elastomer Matrix and Fibres with Significant Bending Stiffness

Fedorova, Svitlana January 2018 (has links)
Constitutive modelling of fibre reinforced solids is the focus of this work. To account for the resulting anisotropy of material, the corresponding strain energy function contains additional terms. Thus, tensile stiffness in the fibre direction is characterised by additional strain invariant and respective material constant. In this way deformation in the fibre direction is penalised. Following this logic, the model investigated in this work includes the term that penalises change in curvature in the fibre direction. The model is based on the large strain anisotropic formulation involving couple stresses, also referred to as “polar elasticity for fibre reinforced solids”. The need of such formulation arises when the size effect becomes significant. Mechanical tests are carried out to confirm the limits of applicability of the classical elasticity for constitutive description of composites with thick fibres. Classical unimaterial models fail to take into account the size affect of fibres and their bending stiffness contribution. The specific simplified model is chosen, which involves new kinematic quantities related to fibre curvature and the corresponding material stiffness parameters. In particular, additional constant k3 (associated with the fibre bending stiffness) is considered. Within the small strains framework, k3 is analytically linked to the geometric and material properties of the composite and can serve as a parameter augmenting the integral stiffness of the whole plate. The numerical tests using the updated finite element code for couple stress theory confirm the relevance of this approach. An analytical study is also carried out, extending the existing solution by Farhat and Soldatos for the fibre-reinforced plate, by including additional extra moduli into constitutive description. Solution for a pure bending problem is extended analytically for couple stress theory. Size effect of fibres is observed analytically. Verification of the new constitutive model and the updated code is carried out using new exact solution for the anisotropic couple stress continuum with the incompressibility constraint. Perfect agreement is achieved for small strain case. Large strain problem is considered by finite element method only qualitatively. Three cases of kinematic constraints on transversely isotropic material are considered in the last section: incompressibility, inextensibility and the double constraint case. They are compared with a general material formulation in which the independent elastic constants are manipulated in order to converge the solution to the “constraint” formulation solution. The problem of a thick plate under sinusoidal load is used as a test problem. The inclusion of couple stresses and additional bending stiffness constant is considered as well. The scheme of determination of the additional constant d31 is suggested by using mechanical tests combined with the analytical procedure.
27

Influence des propriétés mécaniques des chaussures sur la performance en course à pied d'endurance : analyses à court terme et lors d'une course à pied de durée prolongée / Influence of shoe mechanical features on endurance running performance : short term analyses and during a prolonged running duration

Flores, Nicolas 26 June 2019 (has links)
Tandis que les bénéfices physiologiques de chaussures de référence dans le milieu de la course à pied d’endurance sont montrés dans la littérature scientifique, les effets spécifiques et contrôlés de certaines propriétés mécaniques des chaussures demeurent peu connus. L’objectif général de ce travail de thèse était d’étudier les effets du retour d’énergie des semelles intermédiaire des chaussures et de la raideur en flexion des chaussures sur la performance physiologique et biomécanique en course à pied d'endurance. Que ce soit à court-terme ou lors d’une course à pied prolongée, le coût énergétique métabolique (critère utilisé pour évaluer la performance en course à pied) n’était pas significativement modifié par les propriétés mécaniques testées en moyenne parmi le groupe complet de participants. En revanche, les réponses spécifiques aux participants, à la fois à court-terme et lors d’une course à pied prolongée, ont permis de mettre en évidence des combinaisons de réponses biomécaniques et de caractéristiques intrinsèques aux participants expliquant les variations du coût énergétique métabolique en fonction des propriétés mécaniques chaussantes. Une nouvelle stratégie a notamment été mise en évidence chez les participants bénéficiant de la raideur en flexion des chaussures qui se traduisait par une redistribution descendante des activations musculaires des articulations de la hanche et du genou vers l’articulation de la cheville avec la durée de course. Ce travail de thèse soulignait l’importance de considérer une offre de conception de chaussures adaptées à des groupes de coureurs aux réponses biomécaniques et/ou aux caractéristiques intrinsèques similaires. / While physiological benefits of baseline running racing shoes are shown in the scientific literature, the specific and controlled effects of some shoe mechanical features remain not well known. The main purpose of this work was to study the effects of the midsole energy return and the shoe longitudinal bending stiffness on the physiological and biomechanical performance during endurance running. In both short-term and prolonged running duration, the metabolic energetic cost (criteria used to evaluate the endurance running performance) was not significantly altered by the tested mechanical features in average over the group of participants. The main finding was that the shoe mechanical features induced different effects on the metabolic energetic cost depending on the participants. Taking into account the participant-specific responses (in both short-term and during a prolonged running duration) enabled to highlight combinations of biomechanical responses and intrinsic participant characteristics explaining the variations of the metabolic energetic cost as a function of shoe mechanical features. A novel strategy has been highlighted in participants benefiting from the shoe longitudinal bending stiffness resulting in descendant redistribution of the muscular coordination from the hip and knee joints to the ankle joint with the running duration. This work showed the importance of considering a footwear design offer suitable to groups of runners with similar biomechanical responses and/or intrinsic characteristics.
28

Samverkansbjälklag : En studie om KL-platta med samverkande betong

Nilsson, Ida, Svensson, Dennis January 2020 (has links)
Purpose: The purpose of this study was to investigate whether timber-concrete composite (TCC) floors made of a CLT-deck and casted concrete could be used as a method for increasing the use of wood as a building material. Method: The methods used in this degree project were a literature study, in which a number of laws of construction and different connector systems were studied, as well as an experiment consisting of bending tests on CLT-decks with casted concrete where SFS VB-screws were used as shear connectors. Results: The TCC-beams in the experiment behaved as expected and went to bending failure. The beams displayed an almost invisible slip between the materials despite the low composite action of 40,2 %. In addition, the beams had a higher bending stiffness on average than CLT-beams of the same height would have. Conclusions: Compared to floors made entirely of wood, TCC-floors with CLT add extra mass, stiffness, and better acoustic properties to the construction while maintaining a low floor height. The three main types of connectors used in TCC floors are mechanical, notched and glued-in connectors, where the latter two generates higher composite action and stiffness. Adhesive connection is another method with great potential, but there is still more research needed for this type of connection to be used.
29

Estimation of Static Stiffnesses from Free Boundary Dynamic (FRF) Measurements

Pasha, Hasan G. January 2014 (has links)
No description available.
30

Materialval och konstruktion av hållbara tennisracket i trä / Material selection and construction of sustainable wooden tennis rackets

Lundkvist, Benjamin, Spahija, Flamur January 2022 (has links)
Från slutet på 1800-talet till runt 1970-talet gjordes tennisracket i trä och då tekniken inte var så avancerad var de tunga och böjliga. På 1970-talet började dock andra material som exempelvis kolfiber användas som var styvare och gav ett lättare racket. Över tid har metoderna för tillverkningen i trä fallit i glömska vilket gör att undersökningar som denna är intressanta att utföra. Då kolfiber framställs av råolja kan materialet bli mindre attraktivt både ur ett miljö- och ekonomiskt perspektiv då oljepriserna stiger. På grund av detta har företaget Epok Tennis, som arbetet har utförts ihop med, valt att experimentera och prova sig fram i tillverkningen av racket i trä i jakt efter den mest optimala uppsättningen och konstruktionen. Kunskapsläget är dåligt inom området då studie av denna typen generellt görs på större element för användning i exempelvis bostäder och dessa består då framför allt av gran och furu. Därför är det intressant för Epok Tennis att se inverkan av olika träslag som inte vanligtvis används i balkar av i små dimensioner som är jämförbara med tennisracket.Epok Tennis har undersökt hur träracketen byggdes förr och försökt implementera tekniken moderna racket byggs, detta för att skapa en träprodukt som kan fungera lika bra som en modern racket. De viktigaste parametrarna att ta hänsyn till i arbetet var produkternas vikt, styvhet och hållfasthet. I arbetet undersöktes 21 kors- och parallellimmade balkar bestående av totalt sex nordiska träslag. Träslagen i fråga var ask, björk, bok, ek, lönn och valnöt. Sju av balkarna var korslimmade och 14 parallellimade. De korslimmade elementen varierade mellan ett och två tvärgående skikt. Två av balkarna innehöll även produkter som kolfiberduk och linnetyg vilka inte beaktades i beräkningarna men vars resultat var av intresse att analysera. Syftet med arbetet var att undersöka hur olika kombinationer av olika träfanérskivor påverkarböjstyvheten och hållfastheten för en tennisrackets konstruktion. Målet var sedermera att fastställa vilken av balkarna som har högst böjstyvhetoch hållfasthet samt ge förslag om optimal komposituppsättning. Teoretiska beräkningar samt experiment av fyrpunktsböjning utfördes på balkarna där resultaten jämfördes och analyserades. Experimenten utfördes både i balkarnas flat- och kantvisa riktning. Även densiteterna noterades för elementen då konstruktionens totala vikt i slutändan är av stor betydelse.Resultaten av studien visade att de homogena parallellimmade balkarna av björk, lönn och valnöt hade högst böjstyvhet. Vid analys av resultaten noterades att korslimning av elementen generellt försvagade dem i både flat- och kantvis riktning. Till detta ändamål kan det därför vara klokt att endast använda parallellimmade element. I resultatet redovisas även balkarnas knäcklast i kantvisa riktningen.Förekomsten av kvistar i lamellerna ansågs ha en stor inverkan på balkarnas hållfasthet. Därför bör lameller med kvistar försöka undvikas vid framställningen av ramarna till tennisracketen. I förslaget om optimal komposituppsättning har en kombination av lönn och valnöt angetts då dessa visade högst böjstyvhet med avsevärt lägre densitet än björk som dock liknande värde på böjstyvheten.Enligt resultaten av studien är slutsatserna som dragits kortfattat att korslimning generellt försvagar elementen gentemot parallellimning och att bästa komposituppsättning består av lönn och valnöt, i kombination med varandra eller var för sig. / The knowledge of creating wooden tennis rackets has been lost since the emergence of materials like carbon fibre. In this study 21 cross- and parallel laminated beams have been examined, seven of which are cross laminated and 14 parallel laminated. The wood species used were ash, beech, birch, maple, oak and walnut.The purpose of the study was to determine how different combinations of wood lamellas affect strength and stiffness of the beams in terms of bending.The goal was to determine which of the 21 beams had the highest bending stiffness and to give a recommendation of what would be the optimal composite set.Calculations and experiments of four point bending were done in the flat- and edgewise directions, the results of which were compared and analyzed.The results of the study showed that the homogenous parallel laminated beams consisting of birch, maple and walnut had the highest bending stiffness. When analyzing the results, it was found that the parallel laminated beams were generally stronger and stiffer than the cross laminated specimens, for both flat- and edgewise bending.In the recommendation of optimal construction, a combination of maple and walnut has been suggested due to them having similar bending stiffness to birch but significantly lower density.

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