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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.
11

Predictive Modeling For Rate Dependent Toughened-Adhesive Behavior During Impact

Bas, Gamze S. January 2017 (has links)
No description available.
12

Caracterização mecânica e análise de falha de juntas termoplásticas soldadas e termorrígidas coladas de laminados compósitos de grau aeronáutico / Mechanical characterization and failure analysis of thermoplastic welded and thermosetting bonded single-lap joints of aeronautical composite laminates

Castro, Carlos Eduardo Gomes de 23 February 2015 (has links)
Este trabalho objetivou a caracterização mecânica e a análise de falha de dois tipos de juntas compósitas através de ensaios mecânicos, em que uma das juntas era composta por dois laminados de fibras de carbono recoberto em ambas as faces por tecido de fibra de vidro reforçando uma matriz termoplástica (PPS-C) unidos via soldagem por resistência elétrica, e a outra confeccionada com dois laminados de fibras de carbono reforçando uma matriz de resina epóxi (EPX-C) unidos via colagem por filme de resina epoxídica. Os dois tipos de juntas foram submetidos a impacto único transversal de 10 J, condicionamento higrotérmico, além de carregamento em fadiga compressiva no plano nas mais diversas combinações destes processos degradativos de suas propriedades mecânicas. Observou-se, que a junta termorrígida colada de EPX-C apresentou a maior resistência mecânica em flexão em quatro pontos (F4P) na condição original (como-manufaturada), assim como os maiores valores de resistência residual para as várias condições de degradação mecânica e higrotérmica a que foi submetida. Por sua vez, a junta termoplástica soldada de PPS-C exibiu, em termos percentuais, menores reduções dos valores de resistência à flexão sob as condições avaliadas, relativamente às perdas apresentadas pela junta EPX-C em idênticas circunstâncias. A análise macroscópica da superfície de fratura de ambas as juntas indicou que o modo de falha predominante da junta termorrígida foi interfacial enquanto que, para a junta PPS-C, o modo de falha predominante foi o intralaminar. Análises fratográficas através da microscopia eletrônica de varredura (MEV) evidenciaram para a junta termorrígida EPX-C, uma alta adesão entre fibra/matriz, porém uma relativamente fraca interação entre os aderentes (laminado) e o filme adesivo de colagem, enquanto que, para a junta termoplástica PPS-C, reduzidas interações fibra/matriz forem inferidas na camada externa de PPS-V do aderente assim como entre a malha metálica resistiva e os filmes puros de PPS que a revestiam. Em ensaios de resistência ao cisalhamento interlaminar (ILSS), os espécimes retirados da junta EPX-C na condição virgem evidenciaram uma colagem uniforme/homogênea, enquanto que, para a junta PPS-C, os espécimes de ensaio usinados a partir da junta virgem indicaram a ocorrência de efeitos de degradação térmica altamente localizada nas bordas soldadas. / This study aimed to mechanical characterization and failure analysis of two kinds of composite single-lap joints by mechanical tests, in which a single-lap joint was made of two adherents of carbon fibers coated on both sides with glass fiber fabric reinforcing a thermoplastic matrix (PPS-C) welded via resistance welding, and the another type made of two adherents of carbon fibers reinforcing an epoxy matrix (EPX-C) bonded by epoxy adhesive film. The two types of single-lap joints were subjected to single transverse impact of 10 J, hygrothermal conditioning, and compressive fatigue loading in the plane in various combinations of these degradative processes of mechanical properties. It was observed that the thermosetting bonded joint EPX-C showed the highest strength in four point-flexure test in the original condition (as-manufactured), as well as greater residual strength values for the various conditions of mechanical and hygrothermal degradation which was subjected. In turn, the welded thermoplastic joint PPS-C exhibited, in percentage terms, smaller reductions in flexural strength values under the tested conditions, in respect of losses showed by EPX-C in similar circumstances. Macroscopic analysis of the fracture surfaces from both joints indicated that the predominant failure mode was interfacial to thermosetting while for PPS-C joint, the predominant failure mode was intralaminar. Analysis of fracture surfaces by scanning electron microscopy (SEM) showed, for thermosetting joints EPX-C, a high adhesion between fiber/matrix, but a relatively weak interaction between adherents and the bonding adhesive film, whereas for thermoplastic joint PPS-C, reduced interactions fiber/matrix are inferred in the outer layers of PPS-V and between metal mesh and the pure PPS films that coated it. In the interlaminar shear strength tests (ILSS), the specimens removed from the EPX-C joint in the pristine condition showed a uniform/homogeneous bonding along the joint area, while for PPS-C joint, coupons extracted from pristine condition joint indicated the occurrence of degradation caused by thermal effects localized in the welded edges.
13

Detecção por raios-x de trincas de fadiga em juntas rebitadas de Glare&reg / X-ray inspection of fatigue cracks in riveted lap joints of Glare&trade

Soares, Henrique Nogueira 27 April 2007 (has links)
Foi realizado um estudo comparativo da capacidade de duas variantes do método de radiografia por raios- X em detectar trincas de fadiga em juntas sobrepostas rebitadas de laminado híbrido metal-fibra Glare® de uso aeronáutico. Durante os ensaios mecânicos de fadiga sob amplitude constante de tensão, diversos corpos de prova rebitados foram periodicamente inspecionados por raios-X, em ambas as modalidades convencional e digital. Raios-X em filmes convencionais digital. Raios-X em filmes convencionais proporcionaram ótimo detalhamento de trincas nas juntas mecânicas, enquanto que as imagens geradas pela modalidade digital apresentaram qualidade muito inferior, prejudicando ou até mesmo impedindo a avaliação do grau de integridade estrutural das juntas rebitadas. / A comparison is performed on the ability of two modalities of nondestructive X-ray radiography method in detecting fatigue cracks in riveted lap joints of aeronautical grade fiber-metal laminate Glare™. During constant amplitude loading in fatigue testing riveted specimens were periodically inspected using both the conventional and digital X-ray methodology. Conventional film X-ray modality provided high quality images of growing cracks in the mechanical joints, whereas digital radiography generated faulty images, which impaired or even prevented the structural integrity assessment of the riveted joints.
14

Computational modeling, stochastic and experimental analysis with thermoelastic stress analysis for fiber reinforced polymeric composite material systems

Johnson, Shane Miguel 05 May 2010 (has links)
Many studies with Thermoelastic Stress Analysis (TSA) and Infrared Thermography, in Fiber Reinforced Polymeric materials (FRPs), are concerned with surface detection of "hot spots" in order to locate and infer damage. Such experimental analyses usually yield qualitative relations where correlations between stress state and damage severity cannot be obtained. This study introduces quantitative experimental methodologies for TSA and Digital Image Correlation to expand the use of remote sensing technologies for static behavior, static damage initiation detection, and fatigue damage in FRPs. Three major experimental studies are conducted and coupled with nonlinear anisotropic material modeling: static and TSA of hybrid bio-composite material systems, a new stochastic model for fatigue damage of FRPs, and fracture analysis for FRP single-lap joints. Experimental calibration techniques are developed to validate the proposed macromechanical and micromechanical nonlinear anisotropic modeling frameworks under multi-axial states of stress. The High Fidelity Generalized Method of Cells (HFGMC) is a sophisticated micromechanical model developed for analysis of multi-phase composites with nonlinear elastic and elastoplastic constituents is employed in this study to analyze hybrid bio-composites. Macro-mechanical nonlinear anisotropic models and a linear orthotropic model for fracture behavior using the Extended Finite Element method (XFEM) are also considered and compared with the HFGMC method. While micromechanical and FE results provide helpful results for correlating with quasi-static behavior, analyzing damage progression after damage initiation is not straightforward and involves severe energy dissipation, especially with increasing damage progression. This is especially true for fatigue damage evolution, such as that of composite joints as it is associated with uncertainty and randomness. Towards that goal, stochastic Markov Chain fatigue damage models are used to predict cumulative damage with the new damage indices proposed using full-field TSA image analysis algorithms developed for continuously acquired measurements during fatigue loading of S2-Glass/E733FR unidirectional single-lap joints. Static damage initiation is also investigated experimentally with TSA in single-lap joints with thick adherends providing for new design limitations. The computational modeling, stochastic and experimental methods developed in this study have a wide range of applications for static, fracture and fatigue damage of different FRP material and structural systems.
15

Detecção por raios-x de trincas de fadiga em juntas rebitadas de Glare&reg / X-ray inspection of fatigue cracks in riveted lap joints of Glare&trade

Henrique Nogueira Soares 27 April 2007 (has links)
Foi realizado um estudo comparativo da capacidade de duas variantes do método de radiografia por raios- X em detectar trincas de fadiga em juntas sobrepostas rebitadas de laminado híbrido metal-fibra Glare® de uso aeronáutico. Durante os ensaios mecânicos de fadiga sob amplitude constante de tensão, diversos corpos de prova rebitados foram periodicamente inspecionados por raios-X, em ambas as modalidades convencional e digital. Raios-X em filmes convencionais digital. Raios-X em filmes convencionais proporcionaram ótimo detalhamento de trincas nas juntas mecânicas, enquanto que as imagens geradas pela modalidade digital apresentaram qualidade muito inferior, prejudicando ou até mesmo impedindo a avaliação do grau de integridade estrutural das juntas rebitadas. / A comparison is performed on the ability of two modalities of nondestructive X-ray radiography method in detecting fatigue cracks in riveted lap joints of aeronautical grade fiber-metal laminate Glare™. During constant amplitude loading in fatigue testing riveted specimens were periodically inspected using both the conventional and digital X-ray methodology. Conventional film X-ray modality provided high quality images of growing cracks in the mechanical joints, whereas digital radiography generated faulty images, which impaired or even prevented the structural integrity assessment of the riveted joints.
16

Caracterização mecânica e análise de falha de juntas termoplásticas soldadas e termorrígidas coladas de laminados compósitos de grau aeronáutico / Mechanical characterization and failure analysis of thermoplastic welded and thermosetting bonded single-lap joints of aeronautical composite laminates

Carlos Eduardo Gomes de Castro 23 February 2015 (has links)
Este trabalho objetivou a caracterização mecânica e a análise de falha de dois tipos de juntas compósitas através de ensaios mecânicos, em que uma das juntas era composta por dois laminados de fibras de carbono recoberto em ambas as faces por tecido de fibra de vidro reforçando uma matriz termoplástica (PPS-C) unidos via soldagem por resistência elétrica, e a outra confeccionada com dois laminados de fibras de carbono reforçando uma matriz de resina epóxi (EPX-C) unidos via colagem por filme de resina epoxídica. Os dois tipos de juntas foram submetidos a impacto único transversal de 10 J, condicionamento higrotérmico, além de carregamento em fadiga compressiva no plano nas mais diversas combinações destes processos degradativos de suas propriedades mecânicas. Observou-se, que a junta termorrígida colada de EPX-C apresentou a maior resistência mecânica em flexão em quatro pontos (F4P) na condição original (como-manufaturada), assim como os maiores valores de resistência residual para as várias condições de degradação mecânica e higrotérmica a que foi submetida. Por sua vez, a junta termoplástica soldada de PPS-C exibiu, em termos percentuais, menores reduções dos valores de resistência à flexão sob as condições avaliadas, relativamente às perdas apresentadas pela junta EPX-C em idênticas circunstâncias. A análise macroscópica da superfície de fratura de ambas as juntas indicou que o modo de falha predominante da junta termorrígida foi interfacial enquanto que, para a junta PPS-C, o modo de falha predominante foi o intralaminar. Análises fratográficas através da microscopia eletrônica de varredura (MEV) evidenciaram para a junta termorrígida EPX-C, uma alta adesão entre fibra/matriz, porém uma relativamente fraca interação entre os aderentes (laminado) e o filme adesivo de colagem, enquanto que, para a junta termoplástica PPS-C, reduzidas interações fibra/matriz forem inferidas na camada externa de PPS-V do aderente assim como entre a malha metálica resistiva e os filmes puros de PPS que a revestiam. Em ensaios de resistência ao cisalhamento interlaminar (ILSS), os espécimes retirados da junta EPX-C na condição virgem evidenciaram uma colagem uniforme/homogênea, enquanto que, para a junta PPS-C, os espécimes de ensaio usinados a partir da junta virgem indicaram a ocorrência de efeitos de degradação térmica altamente localizada nas bordas soldadas. / This study aimed to mechanical characterization and failure analysis of two kinds of composite single-lap joints by mechanical tests, in which a single-lap joint was made of two adherents of carbon fibers coated on both sides with glass fiber fabric reinforcing a thermoplastic matrix (PPS-C) welded via resistance welding, and the another type made of two adherents of carbon fibers reinforcing an epoxy matrix (EPX-C) bonded by epoxy adhesive film. The two types of single-lap joints were subjected to single transverse impact of 10 J, hygrothermal conditioning, and compressive fatigue loading in the plane in various combinations of these degradative processes of mechanical properties. It was observed that the thermosetting bonded joint EPX-C showed the highest strength in four point-flexure test in the original condition (as-manufactured), as well as greater residual strength values for the various conditions of mechanical and hygrothermal degradation which was subjected. In turn, the welded thermoplastic joint PPS-C exhibited, in percentage terms, smaller reductions in flexural strength values under the tested conditions, in respect of losses showed by EPX-C in similar circumstances. Macroscopic analysis of the fracture surfaces from both joints indicated that the predominant failure mode was interfacial to thermosetting while for PPS-C joint, the predominant failure mode was intralaminar. Analysis of fracture surfaces by scanning electron microscopy (SEM) showed, for thermosetting joints EPX-C, a high adhesion between fiber/matrix, but a relatively weak interaction between adherents and the bonding adhesive film, whereas for thermoplastic joint PPS-C, reduced interactions fiber/matrix are inferred in the outer layers of PPS-V and between metal mesh and the pure PPS films that coated it. In the interlaminar shear strength tests (ILSS), the specimens removed from the EPX-C joint in the pristine condition showed a uniform/homogeneous bonding along the joint area, while for PPS-C joint, coupons extracted from pristine condition joint indicated the occurrence of degradation caused by thermal effects localized in the welded edges.
17

The effects of water displacing corrosion preventatives on the fatigue behaviour of mechanically fastened aluminium joints

Dhamari, Ruby Dharma Adji, Aerospace, Civil & Mechanical Engineering, Australian Defence Force Academy, UNSW January 2005 (has links)
Two failure mechanisms in a double lap joint are investigated. Analytical models of net-section and gross-section failure modes are proposed to describe these mechanisms. The effects of lamping force, interference fit, maximum axial load and WDCP on fatigue performance of the joint are included in the models. The effect of WDCP is assumed to give a reduction in friction coefficient. Three types of stress reduction factors are proposed in the net-section failure model to account for these parameters. The stress reduction factors modify stress range that is used in crack growth calculation. If there are no effects of these parameters, the stress reduction factors are equal to one. Two types of fretting stress are introduced in gross-section failure model to describe either sliding contact or incipient sliding contact on faying surface. The fretting stress is combined with body stress to modify stress range. The net-section failure model predicts that fatigue life is increasing as interference fit, clamping force and friction coefficient increase. The gross-section failure model predicts that fatigue life is decreasing as clamping force and friction coefficient increase. Both models predict that fatigue life is decreasing as maximum axial load increases. Transition of the failure mode occurs earlier as friction coefficient and interference fit increase, while it is delayed as maximum axial load increases. A transition parameter is proposed to establish a relationship between the four main parameters. The transition parameter is expressed in a polynomial equation. It gives an optimum combination of the four main parameters in order to achieve relatively higher fatigue life by having gross-section failure mode. Finite element analysis and fatigue testing are performed to validate the models. The finite element and the analytical models show that stress concentration factor at the edge of the hole is decreasing as clamping force increases. The rate of decrease of stress concentration factor is increasing as friction coefficient increases. While stress concentration factor on the faying surface is increasing as clamping force and friction coefficient increase. Fatigue testing reveals that the fatigue life of the joint is in good agreement with the predicted fatigue life of the proposed models.
18

Simulation de l'effet du confinement sur l'endommagement d'un assemblage en alliage d'aluminium / Simulation of the effect of confinement on the damage inside an aluminum alloy lap joint

Le, Thi My Linh 22 November 2013 (has links)
Pour les structures aéronautiques, les zones sensibles à la corrosion sont principalement celles où le plan de protection de la structure peut être mis en défaut, comme à l’intérieur des joints de recouvrement (dénommé «lap joint» en anglais). Les assemblages de tôles métalliques réalisés par recouvrement en utilisant, par exemple le rivetage, peuvent présenter des défauts de contact et donc il peut exister un interstice entre les deux parties de l’assemblage. En général cet interstice est isolé de l’environnement extérieur par un mastic. Mais il y a un risque de piégeage d’humidité plus ou moins concentrée en ions agressifs qui peut provoquer des endommagements ou aggraver ceux initiés sur des défauts des revêtements. Le premier objectif de cette thèse est de développer une méthodologie permettant de reproduire, de manière contrôlée et de façon la plus représentative de cas réels, la corrosion d’un alliage d’aluminium 2024-T3 constituant un joint de recouvrement afin d’effectuer des observations complétées par des mesures chimiques et électrochimiques. Le deuxième objectif est de tester l’application de modèles de transport réactif (en milieu confiné) pour voir s’il est possible de simuler les réponses (chimiques ou électrochimiques) mesurées expérimentalement et donc de prédire sur le long terme l’évolution d’un assemblage. / For aircraft structures, corrosion sensitive areas are mainly those where the protection of the structure may suffer local breakdowns, as encountered inside lap joints. The assemblies of metal plates by overlapping (riveting) may present some defects in contact and consequently a gap can exist between the two parts of the assembly. Typically this gap is isolated from the outside environment by a sealant. But there is a risk of trapping of moisture more or less concentrated in aggressive ions which can induce corrosion or increase the damages initiated on coating defects. The first objective of this thesis is to develop a methodology to reproduce in a controlled manner and in the most representative way regarding actual cases, the corrosion of a lap joint made of aluminum alloy 2024- T3 plates, to perform not only observations but also chemical and electrochemical measurements. The second objective is to test the application of reactive transport models (in confined media) to check if it is possible to simulate the chemical or electrochemical changes recorded during experiments and thus to predict the long-term evolution of lap-joint in humid environment.
19

MIG/MAG brazing with Cold Metal Transfer / MIG/MAG-lödning med Cold Metal Transfer

ALLVAR, MARIE January 2012 (has links)
In the automotive industry a commonly used material is thin steel sheets coated with a thin layer of zinc for corrosion resistance purposes. Welding of this material, with the high temperatures involved, causes problems with zinc burn-off leading to reduced corrosion resistance. The zinc evaporation also causes arc disturbances leading to spatter formation, pores and difficulties achieving good visual weld quality. The joints are in many cases visible or semi-visible (e.g. visible after opening a door) and “A-grade” quality is demanded, meaning no visible discontinuities are allowed. It also implies a smooth and generally appealing appearance of the joint.An alternative to welding is brazing, and laser brazing meets the requirements but the process is associated with high costs. In the national project “LEX-B”, funded by Vinnova, the possibilities of using arc brazing, in particular one MIG/MAG brazing and two TIG brazing processes, for some automotive applications are investigated. This master thesis is connected to the first part of the project and aims at compiling data for making a selection of the most promising process to investigate and optimise further. LEX-B is conducted in cooperation between Swerea KIMAB, Volvo Trucks, Scania CV and University West. The joint of interest is a lap joint between bottom and upper sheets of 1.2 mm and 0.8 mm respectively that represents a joint on the side of a truck cabin. The requirements are visual A-grade quality and tensile shear strength of 300 MPa.In the project the MIG/MAG process Cold Metal Transfer (CMT) was compared to the TIG processes forceTIG and Plasmatron. CMT was experimentally investigated while the results for the other two processes were obtained partly from a previous study and partly from Volvo Trucks where tests were performed simultaneously. A system for measuring data for the process was developed, test specimens were brazed and examined visually and mechanical destructive testing was performed to ensure the tensile shear strength. Parameter studies were done for further process optimisation. / Inom fordonsindustrin är ett ofta använt material stål i tunna ark belagda med zink p.g.a. zinkets korrossionsskyddande egenskaper. Vid svetsning av detta material uppstår problem med zink som förångas vilket leder till minskad korrosionsbeständighet. Förångningen av zink orsakar också störningar av ljusbågen vilket leder till sprut, porer och svårighet att uppnå god visuell svetskvalitet. Fogarna är i många fall synliga eller delvis synliga (t.ex. synlig efter att ha öppnat en dörr) och "A-kvalitet" efterfrågas, vilket innebär att inga synliga diskontinuiteter tillåts. Fogen ska upplevas jämn och tilltalande.Lödning är ett alternativ till svetsning, och laserlödning är en process som uppfyller kraven men är förknippad med höga kostnader. I det nationella projektet "LEX-B", som finansieras av Vinnova, undersöks möjligheterna att använda båglödning, specifikt en MIG/MAG-process och två TIG-processer, för dessa applikationer. Detta examensarbete är anslutet till den första delen av projektet och syftar till ta fram underlag för att välja den mest lovande av dessa processer för vidare utredning och optimering. LEX-B sker i samarbete mellan Swerea KIMAB, Volvo Lastvagnar, Scania CV och Högskolan Väst. Den undersökta fogen är en överlappsfog mellan en undre plåt med 1,2 mm tjocklek och en övre plåt med 0,8 mm tjocklek. Detta representerar en fog på sidan av en lastbilshytt. Kraven är visuell A-kvalitet och en drag-skjuvhållfasthet av 300 MPa.I projektet jämfördes MIG/MAG-processen Cold Metal Transfer (CMT) med TIG-processerna forceTIG och Plasmatron. CMT testades experimentellt medan resultaten för de övriga två processerna erhölls dels från tidigare tester och dels från Volvo Lastvagnar där tester utfördes parallellt. Ett mätsystem for att dokumentera processdata utvecklades och de framställda lödfogarna undersöktes visuellt och med mekanisk förstörande provning för att säkerställa drag-skjuvhållfastheten. Parameterstudier gjordes för vidare optimering av processen.
20

Experimental and Numerical Investigations on the Durability and Fracture Mechanics of the Bonded Systems for Microelectronics Application

Guo, Shu 01 September 2003 (has links)
Water-assisted crack growth at an epoxy/glass interface was measured as a function of applied strain energy release rate, G, and temperature using a wedge test geometry. The specimens consist of two glass plates bonded with a thin layer of proprietary epoxy adhesive. The crack fronts along the epoxy/glass interfaces were measured using an optical stereomicroscope. The relationship between G and the debonding rate, v, can be measured using this method, and the threshold value of strain energy release rate, Gth, can be determined from the measured data. Two types of testing procedures were conducted in this study: ex situ, i.e., pre-conditioned wedge tests and in situ ones, in which wedges were applied before the specimens were submerged into water. A preliminary model was developed based on the thermal activation barrier concept, and allows the prediction of Gth for the temperatures beyond the testing region. Changes in interfacial strain energy release rate caused by thermal residual stresses in a triple-layered specimen were analyzed in Chapter Three. The method is based on linear elastic fracture mechanics and simple beam theory. The curvature of a bimaterial strip was chosen to characterize the residual stress in the specimen, and the strain energy release rate, caused by both tensile and compressive residual stresses in the adhesive, was derived for an asymmetric double cantilever beam (ADCB) geometry. The contribution of the thermal residual and mechanical stress to the global energy release rate was analyzed. The thermally induced energy release rate, GT, is found to be independent of crack length, but is a function of residual stress level and geometric and material parameters of the specimen. The adhesion of films and coatings to rigid substrates is often measured using blister geometries, which are loaded either by an applied pressure or a central shaft. The measurement will be affected if there are residual stresses that make a contribution to the energy release rate. This effect is investigated using analytical solutions based on the principle of virtual displacements. A geometrically nonlinear finite element analysis is conducted for comparison. Furthermore, the relationships among strain energy release rate, load, deflection, and fracture radius are discussed in detail in Chapter Four. Both analytical solutions and numerical results reveal that uniform tensile residual stresses reduce a specimen's deflection if it experiences plate behavior under small loads. However, this effect diminishes when membrane behavior is dominant. The mechanics of a single-lap joint with different boundary conditions subjected to tensile loading are investigated. Closed-form solutions are obtained for a specimen configuration considering different clamping methods. Based on the approach pioneered by Goland and Reissner, the solutions reported in this paper provide a simple but useful way to understand the effects of boundary conditions on this test geometry. The solutions in this study suggest that different grip configurations mainly affect the response of the specimens if the grip position is close to the joint edge or the loads are small. Generally, the influence caused by different gripping methods is only limited to the boundary region, and the behavior of the joint part subjected to tensile loading is almost the same as that for a simply-supported case. / Ph. D.

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