Search results for: Composite Tubular

Authors: Su-Young Jeong, Won-Kee Hong, Seon-Chee Park, Gyun-Taek Lim, Eric Kim

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Researchers investigate arious strategies to develop composite beams and maximize the structural advantages. This study attempted to conduct experiments and analysis of changes in the neutral axis of positive moments of a Green Beam. Strain compatibility analysis was used, and its efficiency was demonstrated by comparing experimental and analytical values. In the comparison of neutral axis, the difference between experimental and analytical values was found to range from 8.8~26.2%. It was determined that strain compatibility analysis can be useful for predicting the behaviors of composite beams, with the ability to predict the behavior of not only the elastic location of the composite member, but also of the plastic location

Keywords: Composite beam, Strain compatibility, Neutral axis, Green Beam

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Authors: Abdulrahman A. Alghamdi, Paul M. Mummery, Mohammad A. Sheikh

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In this paper, 3D image based composite unit cell is constructed from high resolution tomographic images. Through-thickness thermal diffusivity and in-plane Young’s modulus are predicted for the composite unit cell. The accuracy of the image based composite unit cell is tested by comparing its results with the experimental results obtained from laser flash and tensile test. The FE predictions are in close agreement with experimental results. Through-thickness thermal diffusivity and in-plane Young’s modulus of a virgin C/C composite are predicted by replacing the properties of air (porosity) with the properties of carbon matrix. The effect of porosity was found to be more profound on thermal diffusivity than young’s modulus.

Keywords: Porosity, C/C composite, image based FE modelling, CMC.

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Authors: B. Noruziaan, A. Shvarzman, R. Leahy

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An innovative flooring underlayment was produced and tested. The composite system is made of common OSB boards and a layer of eco-friendly non-cement gypsum based material (GeoGypTM). It was found that the shear bond between the two materials is sufficient to secure the composite interaction between the two. The very high compressive strength and relatively high tensile strength of the non-cement based component together with its high modulus of elasticity provides enough strength and stiffness for the composite product to cover wider spacing between the joists. The initial findings of this study indicate that with joist spacing as wide as 800 mm, the flooring system provides enough strength without compromising the serviceability requirements of the building codes.

Keywords: Composite, floor deck, gypsum based, lumber joist, non-cement, oriented strandboard, shear bond.

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Authors: N. Bahiyah Baba, W. Waugh, A.M. Davidson

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The paper discusses optimising work on a method of processing ceramic / metal composite coatings for various applications and is based on preliminary work on processing anodes for solid oxide fuel cells (SOFCs). The composite coating is manufactured by the electroless co-deposition of nickel and yttria stabilised zirconia (YSZ) simultaneously on to a ceramic substrate. The effect on coating characteristics of substrate surface treatments and electroless nickel bath parameters such as pH and agitation methods are also investigated. Characterisation of the resulting deposit by scanning electron microscopy (SEM) and energy dispersive X-ray analysis (EDXA) is also discussed.

Keywords: Electroless deposition, nickel, YSZ, composite

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Authors: H. Ucar, U. Aridogan

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Composite structures become intriguing for the design of aerospace, automotive and marine applications due to weight reduction, corrosion resistance and radar signature reduction demands and requirements. Studies on piezoelectric ceramic transducers (PZT) for diagnostics and health monitoring have gained attention for their sensing capabilities, however PZT structures are prone to fail in case of heavy operational loads. In this paper, we develop a piezo-based Glass Fiber Reinforced Polymer (GFRP) composite finite element (FE) model, validate with experimental setup, and identify the applicability and limitations of PZTs for a marine application. A case study is conducted to assess the piezo-based sensing capabilities in a representative marine composite structure. A FE model of the composite structure combined with PZT patches is developed, afterwards the response and functionality are investigated according to the sea conditions. Results of this study clearly indicate the blockers and critical aspects towards industrialization and wide-range use of PZTs for marine composite applications.

Keywords: FRP, marine composite, piezoelectric transducer, sea state, wave-induced loads.

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Authors: V. Sadeghi Balkanlou, M. Reza Bagerzadeh Karimi, A. Hasanbakloo, B. Bagheri Azar

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In this paper, the behavior of different types of DST columns has been studied under bending load. Briefly, composite columns consist of an internal carbon steel tube and an external stainless steel wall that the between the walls are filled with concrete. Composite columns are expected to combine the advantages of all three materials and have the advantage of high flexural stiffness of CFDST columns. In this research, ABAQUS software is used for finite element analysis then the results of ultimate strength of the composite sections are illustrated.

Keywords: DST, Stainless steel, carbon steel, ABAQUS, Straigh Columns, Tapered Columns.

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Authors: Elfetori F. Abdewi

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This paper presents the effect of corrugation profile geometry on the crushing behavior, energy absorption, failure mechanism, and failure mode of woven roving glass fibre/epoxy laminated composite tube. Experimental investigations were carried out on composite tubes with three different profile shapes: sinusoidal, triangular and trapezoidal. The tubes were subjected to lateral compressive loading. On the addition to a radial corrugated composite tube, cylindrical composite tube, were fabricated and tested under the same condition in order to know the effect of corrugation geometry. Typical histories of their deformation are presented. Behavior of tubes as regards the peak crushing load, energy absorbed and mode of crushing has been discussed. The results show that the behavior of the tube under lateral compression load is influenced by the geometry of the tube itself.

Keywords: Corrugated composite specimens, Energy absorption, Lateral crushing.

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Authors: Z.A. Rasid, R. Zahari, A. Ayob, D.L. Majid, A.S.M. Rafie

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Aerospace vehicles are subjected to non-uniform thermal loading that may cause thermal buckling. A study was conducted on the thermal post-buckling of shape memory alloy composite plates subjected to the non-uniform tent-like temperature field. The shape memory alloy wires were embedded within the laminated composite plates to add recovery stress to the plates. The non-linear finite element model that considered the recovery stress of the shape memory alloy and temperature dependent properties of the shape memory alloy and composite matrix along with its source codes were developed. It was found that the post-buckling paths of the shape memory alloy composite plates subjected to various tentlike temperature fields were stable within the studied temperature range. The addition of shape memory alloy wires to the composite plates was found to significantly improve the post-buckling behavior of laminated composite plates under non-uniform temperature distribution.

Keywords: Post-buckling, shape memory alloy, temperaturedependent property, tent-like temperature distribution

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Authors: Salem Ali Jebreil

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In this work, a polyaniline/Iron oxide (PANI/Fe2O3) composite was chemically prepared by oxidative polymerization of aniline in acid medium, in presence of ammonium persulphate as an oxidant and amount of Fe2O3. The composite was characterized by a scanning electron microscopy (SEM). The prepared composite has been used as adsorbent to remove Tartrazine dye form aqueous solutions. The effects of initial dye concentration and temperature on the adsorption capacity of PANI/Fe2O3 for Tartrazine dye have been studied in this paper. The Langmuir and Freundlich adsorption models have been used for the mathematical description of adsorption equilibrium data. The best fit is obtained using the Freundlich isotherm with an R2 value of 0.998. The change of Gibbs energy, enthalpy, and entropy of adsorption has been also evaluated for the adsorption of Tartrazine onto PANI/ Fe2O3. It has been proved according the results that the adsorption process is endothermic in nature.

Keywords: Adsorption, Composite, dye, Polyaniline, Tartrazine.

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Authors: Salma Siddika, Fayeka Mansura, Mahbub Hasan

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The term hybrid composite refers to the composite containing more than one type of fiber material as reinforcing fillers. It has become attractive structural material due to the ability of providing better combination of properties with respect to single fiber containing composite. The eco-friendly nature as well as processing advantage, light weight and low cost have enhanced the attraction and interest of natural fiber reinforced composite. The objective of present research is to study the mechanical properties of jute-coir fiber reinforced hybrid polypropylene (PP) composite according to filler loading variation. In the present work composites were manufactured by using hot press machine at four levels of fiber loading (5, 10, 15 and 20 wt %). Jute and coir fibers were utilized at a ratio of (1:1) during composite manufacturing. Tensile, flexural, impact and hardness tests were conducted for mechanical characterization. Tensile test of composite showed a decreasing trend of tensile strength and increasing trend of the Young-s modulus with increasing fiber content. During flexural, impact and hardness tests, the flexural strength, flexural modulus, impact strength and hardness were found to be increased with increasing fiber loading. Based on the fiber loading used in this study, 20% fiber reinforced composite resulted the best set of mechanical properties.

Keywords: Mechanical Properties; Coir, Jute, Polypropylene, Hybrid Composite.

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Authors: Williams S. Ebhota, Freddie L. Inambao

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This study investigates the use of centrifugal casting method to fabricate functionally graded aluminium A356 Alloy and A356-10%SiCp composite for hydro turbine bucket application. The study includes the design and fabrication of a permanent mould. The mould was put into use and the buckets of A356 Alloy and A356-10%SiCp composite were cast, cut and machined into specimens. Some specimens were given T6 heat treatment and the specimens were prepared for different examinations accordingly. The SiCp particles were found to be more at inner periphery of the bucket. The maximum hardness of As-Cast A356 and A356-10%SiCp composite was recorded at the inner periphery to be 60 BRN and 95BRN, respectively. And these values were appreciated to 98BRN and 122BRN for A356 alloy and A356-10%SiCp composite, respectively. It was observed that the ultimate tensile stress and yield tensile stress prediction curves show the same trend.

Keywords: A356 alloy, A356-10%SiCp composite, centrifugal casting, pelton bucket, turbine blade.

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Authors: Sh. Minapoor, S. Ajeli, M. Javadi Toghchi

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Non-crimp 3D orthogonal fabric composite is one of the textile-based composite materials that are rapidly developing light-weight engineering materials. The present paper focuses on geometric and micromechanical modeling of non-crimp 3D orthogonal carbon fabric and composites reinforced with it for aerospace applications. In this research meso-finite element (FE) modeling employs for stress analysis in different load conditions. Since mechanical testing of expensive textile carbon composites with specific application isn't affordable, simulation composite in a virtual environment is a helpful way to investigate its mechanical properties in different conditions.

Keywords: 3D orthogonal woven composite, Aerospace applications, Finite element method, Mechanical properties.

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Authors: Hakim S. Sultan Aljibori

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Effect of geometry on crushing behavior, energy absorption and failure mode of woven roving jute fiber/epoxy laminated composite tubes were experimentally studied. Investigations were carried out on three different geometrical types of composite tubes (circular, square and radial corrugated) subjected to axial compressive loading. It was observed in axial crushing study that the load bearing capability is significantly influenced by corrugation geometry. The influence of geometries of specimens was supported by the plotted load – displacement curves of the tests.

Keywords: Crushing behavior, jute fiber, composite tubes andSpecific energy absorption

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Authors: Onur Ayaz, Nuray Ucar, Elif Bahar, Oguzhan Ucar, Mustafa Oksuz, Aysen Onen, Mehmet Ucar, Ezgi İşmar, Ali Demir

Abstract:

In this study, single nozzle method used for electrospinning technique which composite polymer solution with cellulose nanowiskers (CNW) was treated by ultrasonic sonificator have been compared with coaxial (double) nozzle method, in terms of mechanical, thermal and morphological properties of composite nanofiber. The effect of water content in composite polymer solution on properties of nanofiber has also been examined. It has been seen that single nozzle method which polymer solution does not contain water has better results than that of coaxial method, in terms of mechanical, thermal and morphological properties of nanofiber. However, it is necessary to make an optimization study on setting condition of ultrasonic treatment to get better dispersion of CNW in composite nanofiber and to get better mechanical and thermal properties

Keywords: cellulose nanowhiskers, coaxial nozzle, composite nanofiber, electrospinning

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Authors: Muhammad Khalil, Nader Abuelfoutouh, Gasser Abdelal, Adrian Murphy

Abstract:

Nowadays, the direct effects of lightning to aircrafts are of great importance because of the massive use of composite materials. In comparison with metallic materials, composites present several weaknesses for lightning strike direct effects. Especially, their low electrical and thermal conductivities lead to severe lightning strike damage. The lightning strike direct effects are burning, heating, magnetic force, sparking and arcing. As the problem is complex, we investigated it gradually. A magnetohydrodynamics (MHD) model is developed to simulate the lightning strikes in order to estimate the damages on the composite materials. Then, a coupled thermal-electrical finite element analysis is used to study the interaction between the lightning arc and the composite laminate and to investigate the material degradation.

Keywords: Composite structures, lightning multiphysics, magnetohydrodynamics, coupled thermal-electrical analysis, thermal plasmas.

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Authors: Chien-Wan Hun, Shao-Fu Chang, Jheng-En Yang, Chien-Chon Chen, Wern-Dare Jheng

Abstract:

This research provides a systematic way to study and better understand double nano-tubular structure of alunina (Al2O3) and titania (TiO2). The TiO2 NT was prepared by immersing Al2O3 template in 0.02 M titanium fluoride (TiF4) solution (pH=3) at 25 °C for 120 min, followed by annealing at 450 °C for 1 h to obtain anatase TiO2 NT in the Al2O3 template. Large-scale development of film for nanotube-based CO2 capture and conversion can potentially result in more efficient energy harvesting. In addition, the production process will be relatively environmentally friendly. The knowledge generated by this research will significantly advance research in the area of Al2O3, TiO2, CaO, and Ca2O3 nano-structure film fabrication and applications for CO2 capture and conversion. This green energy source will potentially reduce reliance on carbon-based energy resources and increase interest in science and engineering careers.

Keywords: Alumina, titania, nano-tubular, film, CO2.

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Authors: R. I. Liban, N. Tayşi

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This paper deals with a nonlinear finite element analysis to examine the behavior up to failure of cantilever composite steel-concrete beams which are prestressed externally. 'Pre-' means stressing the high strength external tendons in the steel beam section before the concrete slab is added. The composite beam contains a concrete slab which is connected together with steel I-beam by means of perfect shear connectors between the concrete slab and the steel beam which is subjected to static loading. A finite element analysis will be done to study the effects of external prestressed tendons on the composite steel-concrete beams by locating the tendons in different locations (profiles). ANSYS version 12.1 computer program is being used to analyze the represented three-dimensional model of the cantilever composite beam. This model gives all these outputs, mainly load-displacement behavior of the cantilever end and in the middle span of the simple support part.

Keywords: Composite steel-concrete beams, external prestressing, finite element analysis, ANSYS.

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Authors: Hyun-Seop Shin, Chin-Hyung Lee, Ki-Tae Park

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In order to apply partially filled steel grid composite deck as the horizontal supporting structure of various kinds of infrastructures, the variation of its flexural strength according to design parameters such as cross and longitudinal bars constituting the steel grid and the type of shear connection is evaluated and compared experimentally. The result shows that the design sensitivity of the deck to the spacing of the cross bars is insignificant in the case of structure with low risk of punching failure or without load distribution problem. By means of shear connection composed by transverse rebar and longitudinal bar without additional shear stud bolts, the complete interaction between steel grid and concrete slab is able to be achieved and the composite deck can develop its bending resistance capacity.

Keywords: bending strength, composite action, shear connection, steel grid composite deck

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Authors: K. S. Lim, S. N. A. Azraai, N. M. Noor, N. Yahaya

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Polymeric composites are being increasingly used as repair material for repairing critical infrastructures such as building, bridge, pressure vessel, piping and pipeline. Technique in repairing damaged pipes is one of the major concerns of pipeline owners. Considerable researches have been carried out on the repair of corroded pipes using composite materials. This article attempts a short review of the subject matter to provide insight into various techniques used in repairing corroded pipes, focusing on a wide range of composite repair systems. These systems including pre-cured layered, flexible wet lay-up, pre-impregnated, split composite sleeve and flexible tape systems. Both advantages and limitations of these repair systems were highlighted. Critical technical aspects have been discussed through the current standards and practices. Research gaps and future study scopes in achieving more effective design philosophy are also presented.

Keywords: Composite materials, pipeline, repair technique, polymers.

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Authors: H. Achache, Y. Madani, A. Benzerdjeb

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The present study is based on the three-dimensional digital analysis by the finite elements method of the mechanical loading effect on the delamination of unidirectional and multidirectional stratified composites. The aim of this work is the determination of the release energy rate G in mode I and the Von Mises equivalent constraint distribution along the damaged area under the influence of several parameters such as the applied load and the delamination size. The results obtained in this study show that the unidirectional composite laminates have better mechanical resistance one the loading line than the multidirectional composite laminates.

Keywords: Delamination, release energy rate, stratified composite, finite element method and ply.

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Authors: Sergio Garcia, Alfredo Trueba, Luis M. Vega, Ernesto Madariaga

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Biofilms adhesion is one of the more important cost of industries plants on wide world, which use to water for cooling heat exchangers or are in contact with water. This study evaluated the effect of Electromagnetic Fields on biofilms in tubular heat exchangers using seawater cooling. The results showed an up to 40% reduction of the biofilm thickness compared to the untreated control tubes. The presence of organic matter was reduced by 75%, the inorganic mater was reduced by 87%, and 53% of the dissolved solids were eliminated. The biofilm thermal conductivity in the treated tube was reduced by 53% as compared to the control tube. The hardness in the effluent during the experimental period was decreased by 18% in the treated tubes compared with control tubes. Our results show that the electromagnetic fields treatment has a great potential in the process of removing biofilms in heat exchanger.

Keywords: Biofilm, heat exchanger, electromagnetic fields, seawater.

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Authors: Ch. Nerl, M. Wimmer, P. Hofer, E. Kaschnitz

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The present article deals with a composite casting process that allows to produce bilayer AlSn6-Al strips based on the technique of horizontal continuous casting. In the first part experimental investigations on the production of a single layer AlSn6 strip are described. Afterwards essential results of basic compound casting trials using simple test specimen are presented to define the thermal conditions required for a metallurgical compound between the alloy AlSn6 and pure aluminium. Subsequently, numerical analyses are described. A finite element model was used to examine a continuous composite casting process. As a result of the simulations the main influencing parameters concerning the thermal conditions within the composite casting region could be pointed out. Finally, basic guidance is given for the design of an appropriate composite mould system.

Keywords: Aluminium alloys, composite casting, compound casting, continuous casting, numerical simulation

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Authors: Xuan Sun, Mingbo Tong

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To understand the factors which affect impact damage on composite structures, particularly the effects of impact position and ribs. In this paper, a finite element model (FEM) of low-velocity impact damage on the composite structure was established via the nonlinear finite element method, combined with the user-defined materials subroutine (VUMAT) of the ABAQUS software. The structural elements chosen for the investigation comprised a series of stiffened composite panels, representative of real aircraft structure. By impacting the panels at different positions relative to the ribs, the effect of relative position of ribs was found out. Then the simulation results and the experiments data were compared. Finally, the factors which affect impact damage on the structures were discussed. The paper was helpful for the design of stiffened composite structures.

Keywords: Stiffened, Low-velocity, Impact, Abaqus, Impact Energy.

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Authors: Peng Hao Wang, Garam Kim, Ronald Sterkenburg

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The applications of composite materials within the aviation industry has been increasing at a rapid pace.  However, the growing applications of composite materials have also led to growing demand for more tooling to support its manufacturing processes. Tooling and tooling maintenance represents a large portion of the composite manufacturing process and cost. Therefore, the industry’s adaptability to new techniques for fabricating high quality tools quickly and inexpensively will play a crucial role in composite material’s growing popularity in the aviation industry. One popular tool fabrication technique currently being developed involves additive manufacturing such as 3D printing. Although additive manufacturing and 3D printing are not entirely new concepts, the technique has been gaining popularity due to its ability to quickly fabricate components, maintain low material waste, and low cost. In this study, a team of Purdue University School of Aviation and Transportation Technology (SATT) faculty and students investigated the effectiveness of a 3D printed composite compression mold. A 3D printed composite compression mold was fabricated by 3D scanning a steel valve cover of an aircraft reciprocating engine. The 3D printed composite compression mold was used to fabricate carbon fiber versions of the aircraft reciprocating engine valve cover. The 3D printed composite compression mold was evaluated for its performance, durability, and dimensional stability while the fabricated carbon fiber valve covers were evaluated for its accuracy and quality. The results and data gathered from this study will determine the effectiveness of the 3D printed composite compression mold in a mass production environment and provide valuable information for future understanding, improvements, and design considerations of 3D printed composite molds.

Keywords: Additive manufacturing, carbon fiber, composite tooling, molds.

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Authors: G. Minak, D. Ghelli, A. Zucchelli

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This paper presents the results of an experimental characterization of a glass fibre-epoxy composite. The behavior of the traditional two-phase composite has been compared with the one of a new three-phase composite where the epoxy matrix was modified by addition of a 3% weight fraction of montmorillonite nano-particles. Two different types of nano-clays, Cloisite® 30B and RXG7000, produced by Southern Clay Products Inc., have been considered. Three-point bending tests, both monotonic and cyclic, were carried out. A strong reduction of the ultimate flexural strength upon nano-modification has been observed in quasi-static tests. Fatigue tests yielded a smaller strength loss. In both quasi-static and fatigue tests a more pronounced tendency to delamination has been noticed in three-phase composites, especially in the case of 30B nano-clay, with respect to the standard two-phase glass fiber composite.

Keywords: Bending fatigue, epoxy resin, glass fiber, montmorillonite.

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Authors: Makara Rith, Young Kyu Kim, Seung Woo Lee

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In heavy duty areas, asphalt pavement constructed as entrance roadway may expose distresses such as cracking and rutting during service life. To mitigate these problems, composite pavement with a roller-compacted concrete base may be a good alternative; however, it should be initially investigated. Structural performances such as fatigue cracking and rut depth may be changed due to variation of some design factors. Therefore, this study focuses on the variation effect of material modulus, layer thickness and loading on composite pavement performances. Stress and strain at the critical location are determined and used as the input of transfer function for corresponding distresses to evaluate the pavement performance. Also, composite pavement satisfying the design criteria may be selected as a design section for heavy duty areas. Consequently, this investigation indicates that composite pavement has the ability to eliminate fatigue cracking in asphalt surfaces and significantly reduce rut depth. In addition, a thick or strong rigid base can significantly reduce rut depth and prolong fatigue life of this layer.

Keywords: Composite pavement, ports, cracking, rutting.

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Authors: Mahdi Fakoor, Seyed Mohammad Navid Ghoreishi

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Composite patching is a common way for reinforcing the cracked pipes and cylinders. The effects of composite patch reinforcement on fracture parameters of a cracked pipe depend on a variety of parameters such as number of layers, angle, thickness, and material of each layer. Therefore, stacking sequence optimization of composite patch becomes crucial for the applications of cracked pipes. In this study, in order to obtain the optimal stacking sequence for a composite patch that has minimum weight and maximum resistance in propagation of cracks, a coupled Multi-Objective Genetic Algorithm (MOGA) and Finite Element Method (FEM) process is proposed. This optimization process has done for longitudinal and transverse semi-elliptical cracks and optimal stacking sequences and Pareto’s front for each kind of cracks are presented. The proposed algorithm is validated against collected results from the existing literature.

Keywords: Multi objective optimization, Pareto front, composite patch, cracked pipe.

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Authors: M. M. Mohammadi Dehnavi, A. De Angelis, M. R. Pecce

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This paper deals with the role of studs in structural response for steel-concrete composite beams. A tri-linear slip-shear strength law is assumed according to literature and codes provisions for developing a finite element (FE) model of a case study of a composite deck. The variation of the strength and ductility of the connection is implemented in the numerical model carrying out nonlinear analyses. The results confirm the utility of the model to evaluate the importance of the studs capacity, ductility and strength, on the global response (ductility and strength) of the structures but also to analyse the trend of slip and shear at interface along the beams.

Keywords: Shear Load, slip, steel-concrete composite bridge, stud connectors.

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Authors: Xia Huang, Wenhui Tang, Banghai Jiang, Xianwen Ran

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In this paper, a plane-strain orthotropic elasto-plastic dynamic constitutive model is established, and with this constitutive model, the thermal shock wave induced by intense pulsed X-ray radiation in cylinder shell composite is simulated by the finite element code, then the properties of thermal shock wave propagation are discussed. The results show that the thermal shock wave exhibit different shapes under the radiation of soft and hard X-ray, and while the composite is radiated along different principal axes, great differences exist in some aspects, such as attenuation of the peak stress value, spallation and so on.

Keywords: anisotropic constitutive model, thermal shock wave, X-ray, cylinder shell composite.

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Authors: M.kayhani, M.Nourouzi, A. Amiri Delooei

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This study presents an exact general solution for steady-state conductive heat transfer in cylindrical composite laminates. Appropriate Fourier transformation has been obtained using Sturm-Liouville theorem. Series coefficients are achieved by solving a set of equations that related to thermal boundary conditions at inner and outer of the cylinder, also related to temperature continuity and heat flux continuity between each layer. The solution of this set of equations are obtained using Thomas algorithm. In this paper, the effect of fibers- angle on temperature distribution of composite laminate is investigated under general boundary conditions. Here, we show that the temperature distribution for any composite laminates is between temperature distribution for laminates with θ = 0° and θ = 90° .

Keywords: exact solution, composite laminate, heat conduction, cylinder, Fourier transformation.

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