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Journal articles on the topic 'Curved Beam'
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1
Wang, Yuquan. "Improved Strategy of Two-Node Curved Beam Element Based on the Same Beam’s Nodes Information." Advances in Materials Science and Engineering 2021 (September 2, 2021): 1–9. http://dx.doi.org/10.1155/2021/2093096.
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The curved beam with a great initial curvature is the typical structure and applied widely in real engineering structures. The common practice in the current literature employs two-node straight beam elements as the elementary members for stress and displacement analysis, which needs a large number of divisions to fit the curved beam shape well and increases computational time greatly. In this paper, we develop an improved accurate two-node curved beam element (IC2) in 3D problems, combining the curved Timoshenko beam theory and the curvature information calculated from the same beam curve. The strategy of calculating the curvature information from the same bean curve in the IC2 beam element and then transferring the curvature information to the two-node straight beam element can greatly enhance the accuracy of the mechanical analysis with no extra calculation burden. We then introduce the finite element implementation of the IC2 beam element and verify by the complex curved beam analysis. By comparison with simulation results from the straight two-node beam element in the MIDAS (S2-MIDAS) and the three-node curved beam element adopted in the ANSYS (C3-ANSYS), the simulation results of the typical quarter arc examples under constant or variable curvature show that the IC2 beam element based on curved beam theory is a combination of efficiency and accuracy. And, it is a good choice for analysis of complex engineering rod structure with large initial curvature.
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Mao, Hancheng, Guangbin Yu, Wei Liu, and Tiantian Xu. "Out-of-Plane Free Vibration and Forced Harmonic Response of a Curved Beam." Shock and Vibration 2020 (December 29, 2020): 1–14. http://dx.doi.org/10.1155/2020/8891585.
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Based on the governing differential equation of out-of-plane curved beam, the wave propagation behavior, free vibration, and transmission properties are presented theoretically in this paper. Firstly, harmonic wave solutions are given to investigate the dispersion relation between frequency and wave number, cut-off frequency, displacement, amplitude ratio, and phase diagram. The frequency spectrum results are obtained to verify the work by Kang and Lee. Furthermore, natural frequencies of the single and composite curved beam are calculated through solving the characteristic equation in the case of free-free, clamped-clamped, and free-clamped boundaries. Finally, the transfer matrices of the out-of-plane curved beam are derived by combining the continuity between the different interfaces. The transmissibility curves of the single and composite curved beam are compared to find the vibration attention band. This work will be valuable to extend the study of the out-of-plane vibration of curved beams.
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Pan, Ke-Qi, and Jin-Yang Liu. "Geometric nonlinear dynamic analysis of curved beams using curved beam element." Acta Mechanica Sinica 27, no. 6 (November 18, 2011): 1023–33. http://dx.doi.org/10.1007/s10409-011-0509-x.
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Nadi, Azin, and Mehdi Raghebi. "Finite element model of circularly curved Timoshenko beam for in-plane vibration analysis." FME Transactions 49, no. 3 (2021): 615–26. http://dx.doi.org/10.5937/fme2103615n.
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Curved beams are used so much in the arches and railway bridges and equipments for amusement parks. There are few reports about the curved beam with the effects of both the shear deformation and rotary inertias. In this paper, a new finite element model investigates to analyze In-Plane vibration of a curved Timoshenko beam. The Stiffness and mass matrices of the curved beam element was obtained from the force-displacement relations and the kinetic energy equations, respectively. Assembly of the elemental property matrices is simple and without need to transformation matrix because of using the local polar coordinate system. The natural frequencies of curved Euler-Bernoulli beam with large thickness are not sufficiently accurate. In this case, using the curved Timoshenko beam element is necessary. Moreover, the influence of vibration absorber is discussed on the natural frequencies of the curved beam.
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Song, Yu Min, and Ding Jun Wu. "Establishment of Vibration Differential Equation and Analysis of Dynamics Characteristics for Curved Beam." Advanced Materials Research 250-253 (May 2011): 1329–33. http://dx.doi.org/10.4028/www.scientific.net/amr.250-253.1329.
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In this paper, a differential segment of curved beam in the vibration state is analyzed, and the internal force equilibrium equations are established, then the vibration differential equations of curved beam are derived by considering Timoshenko’s geometric equations and physical equations. The vibration differential equations derived are similar to the Vlasov’s static differential equation of curved beam. By analyzing the vibration differential equations, some characteristics of vibration are obtained, and ideas of solving the vibration differential equations are also proposed. The vibration differential equations of curved beam can be reduced to those of corresponding straight beams, validating the derived vibration differential equation of curved beam.
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Wang, Guang-Ming, Li Zhu, Xin-Lin Ji, and Wen-Yu Ji. "Finite Beam Element for Curved Steel–Concrete Composite Box Beams Considering Time-Dependent Effect." Materials 13, no. 15 (July 22, 2020): 3253. http://dx.doi.org/10.3390/ma13153253.
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Curved steel–concrete composite box beams are widely used in urban overpasses and ramp bridges. In contrast to straight composite beams, curved composite box beams exhibit complex mechanical behavior with bending–torsion coupling, including constrained torsion, distortion, and interfacial biaxial slip. The shear-lag effect and curvature variation in the radial direction should be taken into account when the beam is sufficiently wide. Additionally, long-term deflection has been observed in curved composite box beams due to the shrinkage and creep effects of the concrete slab. In this paper, an equilibrium equation for a theoretical model of curved composite box beams is proposed according to the virtual work principle. The finite element method is adopted to obtain the element stiffness matrix and nodal load matrix. The age-adjusted effective modulus method is introduced to address the concrete creep effects. This 26-DOF finite beam element model is able to simulate the constrained torsion, distortion, interfacial biaxial slip, shear lag, and time-dependent effects of curved composite box beams and account for curvature variation in the radial direction. An elaborate finite element model of a typical curved composite box beam is established. The correctness and applicability of the proposed finite beam element model is verified by comparing the results from the proposed beam element model to those from the elaborate finite element model. The proposed beam element model is used to analyze the long-term behavior of curved composite box beams. The analysis shows that significant changes in the displacement, stress and shear-lag coefficient occur in the curved composite beams within the first year of loading, after which the variation tendency becomes gradual. Moreover, increases in the central angle and shear connection stiffness both reduce the change rates of displacement and stress with respect to time.
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Khaleel, W. H., A. A. Talal, N. H. Baidaa, K. S. Abdul-Razzaq, and A. A. Dawood. "Previous Research Works on Reinforced Concrete Curved Beams." E3S Web of Conferences 318 (2021): 03011. http://dx.doi.org/10.1051/e3sconf/202131803011.
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The current research work summarizes some previous research works on horizontally curved beams. Because of curvature, torsional effects in the analysis and design should be included. Diameter of ring beam, number of supports, beam width, compressive strength of the concrete, and bearing plate width. Which can be summarized from previous studies is that increasing diameter of ring by about 25-75% decreases the capacity load by about 14-36%, while increasing number of supports by about 33-100%, beam width by about 25-75%, compressive strength of concrete by about 24-76%, and bearing plate width by about 25-75% increases the capacity load by about 62-189%, 25-75%, 24-76%, and 5-16%, respectively due to the beam section increase and/or its properties. Frequently, reinforced concrete deep ring beams exhibit shear failure in a manner similar to straight beams. Strut and tie model (STM) and plastic analysis are useful tools for efficiently analyzing ring or curved deep beams. In addition, the nonlinear three-dimensional finite element modeling is typical for predicting the deep curved beams strength and behavior.
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Li, Xiaofei, Haosen Zhai, and Dongyan Zhao. "Out-of-Plane Dynamic Response of Elliptic Curved Steel Beams Based on the Precise Integration Method." Buildings 13, no. 2 (January 28, 2023): 368. http://dx.doi.org/10.3390/buildings13020368.
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The dynamic response of curved steel beams has long been a research focus in curved bridges. The formula for the dynamic response under a moving load was derived according to the basic principles of the precise integration method. Combined with the necessary conditions of this method, the stiffness matrix of a variable-curvature beam was obtained using matrix inversion, and the mass matrix of the structure was obtained using the concentrated mass method. The dynamic response of the structure was obtained by applying moving loads and masses at different speeds to the curved beam. Finite element simulation and laboratory curved-beam models of the variable-curvature steel beam were established. By comparing the laboratory measurement results against the theoretical data obtained in this study, we propose that our theory has practical engineering significance. It can be used as a theoretical basis for the study of variable curvature steel beam structures and for guiding the construction of curved beams.
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Saji, Ms Ansu P., and Ms Lekshmi Priya R. "Flexural Behaviour of SFRC Curved Deep Beams." International Journal for Research in Applied Science and Engineering Technology 10, no. 7 (July 31, 2022): 574–79. http://dx.doi.org/10.22214/ijraset.2022.45372.
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Abstract: As per IS 456:2000, deep beams can be defined as the structures that having span to depth ratio less than 2 for a simply supported beam and 2.5 for a continuous beam. Also these members are loaded on one face and supported on the opposite face. Uses of curved deep beams are increasing in structures like rounded corners of buildings, circular balconies, water tanks etc. Steel Fiber Reinforced Concrete (SFRC) is a concrete with short, discrete lengths of steel fibers which are randomly dispersed. The load deformation behavior of curved deep beam of different curvatures gives an idea about the effect of curvature on the performance of curved deep beam. The structure that generates comparatively small deformation within the applied load can be considered as relatively safe. This paper illustrates the effect of curvature or central angle on the ultimate load behavior of SFRC curved deep beam and analyzing its flexural behaviour. Steel Fiber Reinforced Concrete with 1% steel fiber is used in the current study. The central subtended angles adopted for the study are 00 , 450 , 600 , 900 , 1200 , and 1800 . As the central subtended angle increases, curvature also increases. The analysis of the structure has been carried out using ANSYS Software
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Koziey, B. L., and F. A. Mirza. "Consistent curved beam element." Computers & Structures 51, no. 6 (January 1994): 643–54. http://dx.doi.org/10.1016/s0045-7949(05)80003-3.
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Ecsedi, István, and Ákos József Lengyel. "Deformation of rotating two-layer curved composite beams." Curved and Layered Structures 6, no. 1 (January 1, 2019): 181–91. http://dx.doi.org/10.1515/cls-2019-0015.
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AbstractAn analytical solution is presented for the determination of the deformation of rotating two-layer composite beams. The direction of axis of rotation is vertical and the speed of rotation is constant. The axis of rotation is in the plane of symmetry of curved beam. The source of the in-plane deformation is the stationary rotation of the curved beam. The plane of the curvature is the symmetry plane of the curved beam for its material, geometrical and supporting properties. Assumed form of the displacement field meets the prescriptions of the classical Euler-Bernoulli beam theory. Examples illustrate the applications of the presented analytical solution.
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Jiao, CY, MZ Cheng, H. Cheng, X. Xiao, and YF Wu. "Comparative study of numerical simulation methods for seismic pounding of adjacent girder of curved girder bridges." Journal of Physics: Conference Series 2158, no. 1 (January 1, 2022): 012027. http://dx.doi.org/10.1088/1742-6596/2158/1/012027.
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Abstract Under seismic action, non-uniform collision will occur between the main beams of curved beam bridge, which may lead to local damage of front beam or rear beam. At present, the research on collision effect is mainly based on straight beam bridge, and there is still a lack of relevant research on seismic collision of curved beam bridge. Taking two adjacent typical curved bridges as examples, this paper establishes contact elements (linear elastic model, Kelvin model and Hertz model) and solid elements (three-dimensional contact friction model) which can fully reflect the physical characteristics of seismic collision of curved bridges. By comparing the numerical simulation methods of seismic collision, the advantages and disadvantages of the existing numerical simulation methods in the seismic response analysis of curved bridges are evaluated. The results show that the calculation results of Kelvin model and three-dimensional contact friction model have the least error and high calculation efficiency, and are suitable for the seismic analysis of curved beam bridges.
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Wan, Ze-Qing, Shi-Rong Li, and Hong-Wei Ma. "Geometrically Nonlinear Analysis of Functionally Graded Timoshenko Curved Beams with Variable Curvatures." Advances in Materials Science and Engineering 2019 (June 9, 2019): 1–10. http://dx.doi.org/10.1155/2019/6204145.
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In this paper, geometrically nonlinear analysis of functionally graded curved beams with variable curvatures based on Timoshenko beam theory is presented. Considering the axial extension and the transversal shear deformation, geometrically nonlinear governing equations for the FGM curved beams with variable curvatures subjected to thermal and mechanical loads are formulated. Material properties of the curved beams are assumed to vary arbitrarily in the thickness direction and be independent on the temperature change. By using the numerical shooting method to solve the coupled ordinary differential equations, the nonlinear response of static thermal bending of a FGM semielliptic beams subjected to transversely nonuniform temperature rise is obtained numerically. The effects of material gradient, shear deformation, and temperature rise on the response of the curved beam are discussed in detail. Nonlinear bending of a closed FGM elliptic structure subjected to two pinching concentrated loads is also analyzed. This paper presents some equilibrium paths and configurations of the elliptic curved beam for different pinching concentrated loads.
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He, Yanfei, Xingwu Zhang, Jia Geng, Xuefeng Chen, and Zengguang Li. "Two Kinds of Finite Element Variables Based on B-Spline Wavelet on Interval for Curved Beam." International Journal of Applied Mechanics 11, no. 02 (March 2019): 1950017. http://dx.doi.org/10.1142/s1758825119500170.
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Curved beam structure has been widely used in engineering, due to its good load-bearing and geometric characteristics. More common methods for analyzing and designing this structure are the finite element methods (FEMs), but these methods have many disadvantages. Fortunately, the multivariable wavelet FEMs can solve these drawbacks. However, the multivariable generalized potential energy functional of curved beam, used to construct this element, has not been given in previous literature. In this paper, the generalized potential energy functional for curved beam with two kinds of variables is derived initially. On this basis, the B-spline wavelet on the interval (BSWI) is used as the interpolation function to construct the wavelet curved beam element with two kinds of variables. In the end, several typical numerical examples of thin to thick curved beams are given, which show that the present element is more effective in static and free vibration analysis of curved beam structures.
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Lasry, Gabriel, Yaniv Brick, and Timor Melamed. "Manipulation of curved beams using beam-domain optimization." Optics Express 30, no. 4 (February 9, 2022): 6061. http://dx.doi.org/10.1364/oe.449871.
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Witwit, Dolfocar, and Nabeel Jasim. "Behaviour of New Curved in Plan Composite Reinforced Concrete Beams." Basrah journal for engineering science 22, no. 2 (December 24, 2022): 80–89. http://dx.doi.org/10.33971/bjes.22.2.12.
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New composite reinforced concrete beams, in which reinforced concrete component is connected to steel T-section, are proposed. The stirrups of the beam were utilized as shear connectors by passing them through drilled holes in the web of the steel T-section. Experimental test and numerical analysis were conducted to determine the behaviour of such beams when subjected to combined shear, torsion, and bending stresses. Full scale one conventional reinforced concrete curved in plan beam C1, and four composite reinforced concrete ones, C2 to C5, were tested. The degree of shear connection between the two components of beams C2 to C5 was changed by varying the number of stirrups which are used as shear connectors. The increase in load carrying capacity of the composite reinforced concrete beams reached 55 % for beam C4 as compared to that of ordinary reinforced concrete beam. The experimental results demonstrated that the stirrups are very effective in providing the interaction between the two components of the beams. The degree of shear connection emerged not to have effect on the behaviour of tested beams. Three-dimensional finite element analysis was conducted using commercial software ABAQUS. To model the shear connection in composite reinforced concrete beam, the stirrups were connected to the web of the steel T-section by springs at the location of the stirrups. Good agreement is obtained between the results of the experimental tests and the finite element analysis.
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Liu, Zhi-Min, Xue-Jin Huo, Guang-Ming Wang, and Wen-Yu Ji. "Test and Numerical Model of Curved Steel–Concrete Composite Box Beams under Positive Moments." Materials 14, no. 11 (May 31, 2021): 2978. http://dx.doi.org/10.3390/ma14112978.
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Compared with straight steel–concrete composite beams, curved composite beams exhibit more complicated mechanical behaviors under combined bending and torsion coupling. There are much fewer experimental studies on curved composite beams than those of straight composite beams. This study aimed to investigate the combined bending and torsion behavior of curved composite beams. This paper presents static loading tests of the full elastoplastic process of three curved composite box beams with various central angles and shear connection degrees. The test results showed that the specimens exhibited notable bending and torsion coupling force characteristics under static loading. The curvature and interface shear connection degree significantly affected the force behavior of the curved composite box beams. The specimens with weak shear connection degrees showed obvious interfacial longitudinal slip and transverse slip. Constraint distortion and torsion behavior caused the strain of the inner side of the structure to be higher than the strain of the outer side. The strain of the steel beam webs was approximately linear. In addition, fine finite element models of three curved composite box beams were established. The correctness and applicability of the finite element models were verified by comparing the test results and numerical calculation results for the load–displacement curve, load–rotational angle curve, load–interface slip curve, and cross-sectional strain distribution. Finite element modeling can be used as a reliable numerical tool for the large-scale parameter analysis of the elastic–plastic mechanical behavior of curved composite box beams.
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Dvořáková, Edita, and Bořek Patzák. "ON COMPARISON OF 3D ISOGEOMETRIC TIMOSHENKO AND BERNOULLI BEAM FORMULATIONS." Acta Polytechnica CTU Proceedings 30 (April 22, 2021): 12–17. http://dx.doi.org/10.14311/app.2021.30.0012.
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Application of isogeometric analysis (IGA) for curved beams is very convenient for its ability of exact representation of curved geometries. Several beam formulation has been presented since the introduction of IGA. In this paper, two different beam formulations are presented: Bernoulli beam formulation of A. M. Bauer et al. [1], and Timoshenko beam element introduced by G. Zhang et al. [2]. Both beam elements are implemented and their performance is documented on the fully threedimensionalexample of helicoidal spring.
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Strozzi, Antonio, Enrico Bertocchi, and Sara Mantovani. "A paradox in curved beams." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 233, no. 8 (September 6, 2018): 2830–33. http://dx.doi.org/10.1177/0954406218797980.
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It is sometimes possible to relieve the stresses in a mechanical component by removing material, where relief grooves are the commonest expedient approach. Within the rectilinear beam realm, rare situations are known in which, by removing material in the cross-sectional zones that are farthest from the neutral axis, a bending stress diminution is achieved. With regard to curved beams, selected examples are presented in which a bending stress diminution is achieved by laterally removing material from the zones close to the neutral axis. An approximate mathematical approach based on Gateaux linearization is developed that delimits the lateral zones of the beam cross-section in which material removal is accompanied by bending stress reduction. While the achievable stress diminution is generally marginal, the reduction of the beam’s cross-section is technically interesting.
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Leung, A. Y. T., and W. E. Zhou. "Dynamic Stiffness Analysis of Curved Thin-Walled Beams." Shock and Vibration 1, no. 1 (1993): 77–88. http://dx.doi.org/10.1155/1993/374730.
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The natural vibration problem of curved thin-walled beams is solved by the dynamic stiffness method. The dynamic stiffness of a curved open thin-walled beam is given. The computed natural frequencies of the beam are compared with those obtained by a completely analytical method to show the high accuracy of the present method. The interaction of in-plane and out-of-plane modes is emphasized.
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Cammarata, Alessandro, Pietro Davide Maddio, Rosario Sinatra, and Nicola Pio Belfiore. "Direct Kinetostatic Analysis of a Gripper with Curved Flexures." Micromachines 13, no. 12 (December 8, 2022): 2172. http://dx.doi.org/10.3390/mi13122172.
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Micro-electro-mechanical-systems (MEMS) extensively employed planar mechanisms with elastic curved beams. However, using a curved circular beam as a flexure hinge, in most cases, needs a more sophisticated kinetostatic model than the conventional planar flexures. An elastic curved beam generally allows its outer sections to experience full plane mobility with three degrees of freedom, making complex non-linear models necessary to predict their behavior. This paper describes the direct kinetostatic analysis of a planar gripper with an elastic curved beam is described and then solved by calculating the tangent stiffness matrix in closed form. Two simplified models and different contributions to derive their tangent stiffness matrices are considered. Then, the Newton–Raphson iterative method solves the non-linear direct kinetostatic problem. The technique, which appears particularly useful for real-time applications, is finally applied to a case study consisting of a four-bar linkage gripper with elastic curved beam joints that can be used in real-time grasping operations at the microscale.
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Dvořáková, Edita, and Bořek Patzák. "Locking Removal Techniques for the Isogeometric Formulation of Curved Beams." Advanced Materials Research 1144 (March 2017): 109–14. http://dx.doi.org/10.4028/www.scientific.net/amr.1144.109.
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The isogeometric formulation seems to be advantageous especially when it comes to curved geometries such as curved beams and shells. In this paper, the NURBS isogeometric formulation of beam element is presented. The same basis functions are used for both geometry description and unknown approximations so there is no accuracy loss caused by a geometry approximation. The element is based on Timoshenko beam theory which enables the use of the element for both thick and thin beams, nevertheless in case of thin beams the shear-locking phenomena is observed. In the paper it is shown that the reduced integration is insufficient for locking removal and capability of Discrete Shear Gap (DSG) method to unlock the elements is examined. For clear demonstration of locking-removal techniques the implemented element is first tested for the case of straight beam, then the performance is demonstrated on the curved geometry.
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Rajasekaran, S. "Analysis of curved beams using a new differential transformation based curved beam element." Meccanica 49, no. 4 (November 20, 2013): 863–86. http://dx.doi.org/10.1007/s11012-013-9835-3.
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Festersen, Sven, Stjepan B. Hrkac, Christian T. Koops, Benjamin Runge, Thomas Dane, Bridget M. Murphy, and Olaf M. Magnussen. "X-ray reflectivity from curved liquid interfaces." Journal of Synchrotron Radiation 25, no. 2 (February 16, 2018): 432–38. http://dx.doi.org/10.1107/s1600577517018057.
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X-ray reflectivity studies of the structure of liquid–vapour and liquid–liquid interfaces at modern sources, such as free-electron lasers, are currently impeded by the lack of dedicated liquid surface diffractometers. It is shown that this obstacle can be overcome by an alternative experimental approach that uses the natural curvature of a liquid drop for variation of the angle of incidence. Two modes of operation are shown: (i) sequential reflectivity measurements by a nanometre beam and (ii) parallel acquisition of large ranges of a reflectivity curve by micrometre beams. The feasibility of the two methods is demonstrated by studies of the Hg/vapour, H2O/vapour and Hg/0.1 MNaF interface. The obtained reflectivity curves match the data obtained by conventional techniques up to 5αcin micro-beam mode and up to 35αcin nano-beam mode, allowing observation of the Hg layering peak.
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Sun, Jian Peng, and Qing Ning Li. "The Precise Transfer Matrix Method for Dynamic Characteristic Analysis of Curved Box Beams." Advanced Materials Research 255-260 (May 2011): 1721–24. http://dx.doi.org/10.4028/www.scientific.net/amr.255-260.1721.
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The dynamic characteristic as one key feature determines the earthquake response and seismic performance of structures.Curved box beams occur coupling of flexural and torsional modes,which companied with warping.Based on the precise transfer matrix method,precise transfer matrixs for solving the natural frequencies and modes of curved box beam out plane have been derivated.The analysis of vibration characteristics of single-span curved box beam with pinned-pinned ends has been done. Example shows that the precise transfer matrix method is a simple and effective method for dynamic characteristic analysis of curved box beams.
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Ghuku, Sushanta, and Kashi Nath Saha. "A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection." International Journal of Engineering and Technologies 11 (July 2017): 13–39. http://dx.doi.org/10.18052/www.scipress.com/ijet.11.13.
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The paper presents a review on large deflection behavior of curved beams, as manifested through the responses under static loading. The term large deflection behavior refers to the inherent nonlinearity present in the analysis of such beam system response. The analysis leads to the field of geometric nonlinearity, in which equation of equilibrium is generally written in deformed configuration. Hence the domain of large deflection analysis treats beam of any initial configuration as curved beam. The term curved designates the geometry of center line of beam, distinguishing it from the usual straight or circular arc configuration. Different methods adopted by researchers, to analyze large deflection behavior of beam bending, have been taken into consideration. The methods have been categorized based on their application in various formats of problems. The nonlinear response of a beam under static loading is also a function of different parameters of the particular problem. These include boundary condition, loading pattern, initial geometry of the beam, etc. In addition, another class of nonlinearity is commonly encountered in structural analysis, which is associated with nonlinear stress-strain relations and known as material nonlinearity. However the present paper mainly focuses on geometric nonlinear analysis of beam, and analysis associated with nonlinear material behavior is covered briefly as it belongs to another class of study. Research works on bifurcation instability and vibration responses of curved beams under large deflection is also excluded from the scope of the present review paper.
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Ghuku, Sushanta, and Kashi Nath Saha. "A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection." International Journal of Engineering and Technologies 11 (July 13, 2017): 13–39. http://dx.doi.org/10.56431/p-48538j.
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The paper presents a review on large deflection behavior of curved beams, as manifested through the responses under static loading. The term large deflection behavior refers to the inherent nonlinearity present in the analysis of such beam system response. The analysis leads to the field of geometric nonlinearity, in which equation of equilibrium is generally written in deformed configuration. Hence the domain of large deflection analysis treats beam of any initial configuration as curved beam. The term curved designates the geometry of center line of beam, distinguishing it from the usual straight or circular arc configuration. Different methods adopted by researchers, to analyze large deflection behavior of beam bending, have been taken into consideration. The methods have been categorized based on their application in various formats of problems. The nonlinear response of a beam under static loading is also a function of different parameters of the particular problem. These include boundary condition, loading pattern, initial geometry of the beam, etc. In addition, another class of nonlinearity is commonly encountered in structural analysis, which is associated with nonlinear stress-strain relations and known as material nonlinearity. However the present paper mainly focuses on geometric nonlinear analysis of beam, and analysis associated with nonlinear material behavior is covered briefly as it belongs to another class of study. Research works on bifurcation instability and vibration responses of curved beams under large deflection is also excluded from the scope of the present review paper.
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Qiu, J., J. H. Lang, and A. H. Slocum. "A Curved-Beam Bistable Mechanism." Journal of Microelectromechanical Systems 13, no. 2 (April 2004): 137–46. http://dx.doi.org/10.1109/jmems.2004.825308.
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Williams, G. J., H. M. Quiney, B. B. Dahl, C. Q. Tran, A. G. Peele, K. A. Nugent, M. D. De Jonge, and D. Paterson. "Curved beam coherent diffractive imaging." Thin Solid Films 515, no. 14 (May 2007): 5553–56. http://dx.doi.org/10.1016/j.tsf.2006.12.132.
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Choi, Jong Keun, and Jang Keun Lim. "Simple curved shear beam elements." Communications in Numerical Methods in Engineering 9, no. 8 (August 1993): 659–69. http://dx.doi.org/10.1002/cnm.1640090805.
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Özyiğit, Hamdi Alper, Mehmet Yetmez, and Utku Uzun. "Out-of-Plane Vibration of Curved Uniform and Tapered Beams with Additional Mass." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/8178703.
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As there is a gap in literature about out-of-plane vibrations of curved and variable cross-sectioned beams, the aim of this study is to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered. Out-of-plane free vibration of curved uniform and tapered beams with additional mass is also investigated. Finite element method is used for all analyses. Curvature type is assumed to be circular. For the different boundary conditions, natural frequencies of both symmetrical and unsymmetrical tapered beams are given together with that of uniform tapered beam. Bending, torsional, and rotary inertia effects are considered with respect to no-shear effect. Variations of natural frequencies with additional mass and the mass location are examined. Results are given in tabular form. It is concluded that (i) for the uniform tapered beam there is a good agreement between the results of this study and that of literature and (ii) for the symmetrical curved tapered beam there is also a good agreement between the results of this study and that of a finite element model by using MSC.Marc. Results of out-of-plane free vibration of symmetrically tapered beams for specified boundary conditions are addressed.
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Hao, Ying, Wei He, and Yanke Shi. "Differential Equations of Motion for Naturally Curved and Twisted Composite Space Beams." Shock and Vibration 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/5015807.
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The differential equations of motion for naturally curved and twisted elastic space beams made of anisotropic materials with noncircular cross sections, being a coupled system consisting of 14 second-order partial differential equations with variable coefficients, are derived theoretically. The warping deformation of beam’s cross section, as a new design factor, is incorporated into the differential equations in addition to the anisotropy of material, the curvatures of the rod axis, the initial twist of the cross section, the rotary inertia, and the shear and axial deformations. Numerical examples show that the effect of warping deformation on the natural frequencies of the beam is significant under certain geometric and boundary conditions. This study focuses on improving and consummating the traditional theories to build a general curve beam theory, thereby providing new scientific research reference and design principle for curve beam designers.
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He, Xiao-Ting, Xin Wang, Meng-Qiao Zhang, and Jun-Yi Sun. "The Thermal Stress Problem of Bimodular Curved Beams under the Action of End-Side Concentrated Shear Force." Materials 16, no. 15 (July 25, 2023): 5221. http://dx.doi.org/10.3390/ma16155221.
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A bimodular material is a kind of material that presents two elastic moduli in tension and compression. In classical thermoelasticity, however, the bimodular material is rarely considered due to its complexity in analysis. In fact, almost all materials will present, more or less, bimodular characteristics, and in some cases, the mechanical properties of materials cannot be fully utilized simply by ignoring the bimodular characteristics. In this study, the thermal stress problem of bimodular curved beams under the action of end-side concentrated shear force is analytically and numerically investigated, in which the temperature rise modes in a thermal environment are considered arbitrary. Using the stress function method based on compatibility conditions, a two-dimensional solution of thermoelasticity of the bimodular curved beam subjected to end-side concentrated shear force was obtained. The results show that the solution for a bimodular curved beam with a thermal effect can be reduced to that of a bimodular curved beam without a thermal effect. At the same time, the numerical simulation for the problem verifies the correctness of the theoretical solution. The results may serve as a theoretical reference for the refined analysis and optimization of curved beams in a thermal environment.
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Hussein, Ahmad Alaa, and Ahmad Jabbar Hussain Alshimmeri. "Comparative Study of Structural Behavior for Asymmetrical Castellated (Concavely - Curved Soffit) Steel Beams with Different Strengthening Techniques." Key Engineering Materials 895 (August 3, 2021): 177–89. http://dx.doi.org/10.4028/www.scientific.net/kem.895.177.
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The Asymmetrical Castellated concavely – curved soffit Steel Beams with RPC and Lacing Reinforcement improves compactness and local buckling (web and flange local buckling), vertical shear strength at gross section (web crippling and web yielding at the fillet), and net section ( net vertical shear strength proportioned between the top and bottom tees relative to their areas (Yielding)), horizontal shear strength in web post (Yielding), web post-buckling strength, overall beam flexure strength, tee Vierendeel bending moment and lateral-torsional buckling, as a result of steel section encasement. This study presents two concentrated loads test results for seven specimens Asymmetrical Castellated concavely – curved soffit Steel Beams section encasement by Reactive powder concrete (RPC) with laced reinforcement. The encasement of the Asymmetrical Castellated concavely – curved soffit Steel Beams consists of, flanges unstiffened element height was filled with RPC for each side, and laced reinforced which are used inclined continuous reinforcement of two layers on each side of the Asymmetrical Castellated concavely – curved soffit Steel Beams web. The inclination angle of lacing reinforcement concerning the longitudinal axis is 45. Seven specimens with seven different configurations will be prepared and tested under two concentrated loads at the mid-third of the beam span. The tested specimen's properties are: unconfined Asymmetrical Castellated Steel Beams (Reference1), second model; Asymmetrical Castellated concavely – curved soffit Steel Beams (web and flange) confined with (RPC) only, third model; Asymmetrical Castellated concavely – curved soffit Steel Beams (web and flange) confined with (RPC) and laced reinforcement, fourth model; is same as the third model but it has one web opening with increase the depth of web post by 10 %, 20%, and 30 % as a gap between top and bottom parts of Asymmetrical Castellated concavely – curved soffit Steel Beams respectively. The results that have been obtained from the experimental part and the numerical analysis results by ABAQUS demonstrated that the increase of the gap leads to an increase in the load against the deflection curve. Sample CB8 with 122 mm gap has gained the highest load against deflection when compared with either reference sample without gap and other samples with 65 mm and 105 mm gap for concavely–curved soffit Steel Beams.
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Baxy, Ajinkya, and Abhijit Sarkar. "Natural frequencies of a rotating curved cantilever beam: A perturbation method-based approach." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 234, no. 9 (January 20, 2020): 1706–19. http://dx.doi.org/10.1177/0954406219899117.
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The blades of propellers, fans, compressor and turbines can be modeled as curved beams. In general, for industrial application, finite element method is employed to determine the modal characteristics of these structures. In the present work, a novel formula for determining the natural frequencies of a rotating circularly curved cantilever beam is derived. Rayleigh–Ritz approach is used along with perturbation method to obtain the analytical formula. In the first part of the work, a formula for natural frequencies of a non-rotating curved beam vibrating in its plane of curvature is presented. This formula is derived as a correction to the natural frequencies of its straight counterpart. The curvature is treated as a perturbation parameter. In the next part of the work, the effect of rotation on the curved beam is captured as an additional perturbation. Thus, the formula for a curved rotating beam is derived as a correction (involving two perturbation parameters) to the non-rotating straight beam. The results obtained using the derived formula are compared with the finite element method results. It is found that the frequency estimates from the formula are valid over a fairly large range of curvature and rotation speed. Thus, the derived formula can provide a faster alternative for design iterations in industrial applications.
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Yang, Zeying, Chenghe Wang, Yinglin Sun, Yangyudong Liu, Zhengquan Cheng, and Weisong Qu. "Out-of-Plane Deformation Analysis of the Thin-Walled Closed Curved Box Girder under the Temperature Gradient." Advances in Civil Engineering 2021 (October 16, 2021): 1–12. http://dx.doi.org/10.1155/2021/8662270.
Full textAbstract:
For calculating the thin-walled closed curved box girder caused by the temperature gradient of the internal force and displacement, based on the fundamental differential equation of the curve beam and the principle of minimum energy, set a reverse statically indeterminate simply supported curve beam as the basic structure, consider the warping effect of the closed curve box girder, and put forward a kind of plane curve beam temperature deformation simple analytical calculation method. Compared with the finite element calculation results, the relative error of the analytical calculation results is less than 5%. It is concluded that the analytical method has sufficient accuracy in calculating the out-of-plane deformation of the thin-walled closed curved box girder under the temperature gradient.
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Khater, Mahmoud E. "A Generalized Model for Curved Nanobeams Incorporating Surface Energy." Micromachines 14, no. 3 (March 16, 2023): 663. http://dx.doi.org/10.3390/mi14030663.
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This work presents a comprehensive model for nanobeams, incorporating beam curvature and surface energy. Gurtin–Murdoch surface stress theory is used, in conjunction with Euler–Bernoulli beam theory, to model the beams and take surface energy effects into consideration. The model was validated by contrasting its outcomes with experimental data published in the literature on the static bending of fixed–fixed and fixed–free nanobeams. The outcomes demonstrated that surface stress alters the stiffness of both fixed–fixed and fixed–free nanobeams with different behaviors in each case.
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Stephen, N. G. "How long is a curved beam?" Journal of Strain Analysis for Engineering Design 40, no. 3 (April 1, 2005): 295–97. http://dx.doi.org/10.1243/030932405x7719.
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For a straight or thin curved beam, the expression for strain energy due to bending is U = M2 L/(2EI); for this to be applicable to a thick curved beam, the requisite length is slightly greater than the centre-line length.
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Liu, Yuechang, Xin Zhang, Jianhua Guo, Hang Yang, Lixiang Han, Yuanwei Yao, and Fugen Wu. "Tailoring of diversified sound vortices using curved impedance-matched acoustic metasurfaces." Modern Physics Letters B 34, no. 12 (February 18, 2020): 2050121. http://dx.doi.org/10.1142/s0217984920501213.
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Acoustic vortex beam, which carries the orbital angular momentum, has attracted great interest in recent years. In this paper, we propose a novel curved impedance-matched metasurface constructed with tunable units, which are filled with mixed gases with different refractive indices. It converts an acoustic point source to vortex beam with nearly unity transmittance in a broad frequency bandwidth. By arranging the units, we demonstrate topological charge modulation and rotated direction shifting for acoustic vortex beams. In addition, combining with phase superposition effect, we introduce double curved metasurfaces model. A vortex beam can transform into another vortex beam with different energy flows conveniently. The proposed methods open up an effective avenue for tailoring acoustic vortex fields and provide possibility for applications such as particles trapping, acoustic communication and biomedical engineering.
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Yang, F., R. Sedaghati, and E. Esmailzadeh. "Free in-plane vibration of curved beam structures: A tutorial and the state of the art." Journal of Vibration and Control 24, no. 12 (August 28, 2017): 2400–2417. http://dx.doi.org/10.1177/1077546317728148.
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The study of free in-plane vibration of curved beams, using different beam theories, is more challenging than that of straight beams, since the structural deformations in curved beams depend not only on the rotation and radial displacements, but also on the coupled tangential displacement caused by the curvature of structures. A critical review of the publications on the free in-plane vibration of curved beams to demonstrate the state of the art has been presented. The governing differential equations of motion for the curved beams, based on different hypotheses (including and excluding the axial extensity, rotary inertia and the shear deformation), were discussed and different approaches to solve the developed equations of motion have been identified. Finally, a systematic comparison of the dynamic properties of curved beams evaluated with various forms of curvatures based on different hypotheses were presented.
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Jaafer, Abdulkhaliq A., and Saba L. Kareem. "Behavior of Curved Steel-Concrete Composite Beams Under Monotonic Load." International Journal of Mathematical, Engineering and Management Sciences 5, no. 6 (December 1, 2020): 1210–33. http://dx.doi.org/10.33889/ijmems.2020.5.6.091.
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The paper develops a numerical investigation on the behavior of steel-concrete composite beam curved in plan to examine the effect of the various parameters. Three-dimensional finite element analysis (FEA) is employed using a commercial software, ABAQUS. The geometric and material nonlinearities are utilized to simulate the composite beam under a monotonic load. The FEA efficiency has been proved by comparing the numerical results with experimental tests obtained from previous literature, including load-deflection curves, ultimate load, ultimate and failure deflection, and cracks propagation. The validated models are used to assess some of the key parameters such the beam span/radius ratio, web stiffeners, partial interaction, concrete compressive strength, and steel beam yield stress. From the obtained results, it is noticed that the span/radius of curvature ratio influences the loading capacity, the beam yielding (i.e. the beam yield at an early stage) when the span/radius ratio increases and inelastic behavior developed early of the beam due to the torsional effect. The presence of web stiffeners with different locations in the curve composite beam affected the shear strength. The web twisting and vertical separation at the beam mid-span are observed to decrease as the number of the stiffeners increase due to the decrease in the beam torsion incorporating with transferring the failure to the concrete slab. Furthermore, the partial interaction and steel beam yield stress developed in this study appear to have a remarkable effect on beam capacity.
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Chidamparam, P., and A. W. Leissa. "Vibrations of Planar Curved Beams, Rings, and Arches." Applied Mechanics Reviews 46, no. 9 (September 1, 1993): 467–83. http://dx.doi.org/10.1115/1.3120374.
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This work attempts to organize and summarize the extensive published literature on the vibrations of curved bars, beams, rings and arches of arbitrary shape which lie in a plane. In-plane, out-of-plane and coupled vibrations are considered. Various theories that have been developed to model curved beam vibration problems are examined. An overview is presented of the types of problems which are addressed in the literature. Particular attention is given to the effects of initial static loading, nonlinear vibrations and the application of finite element techniques. The significantly different frequencies arising from curved beam theories which either allow or prevent extension of the centerline during vibratory motion are shown. An extensive bibliography of 407 relevant references is included.
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ERKMEN, R. EMRE, and MARK A. BRADFORD. "NONLINEAR ELASTO-DYNAMIC ANALYSIS OF I-BEAMS CURVED IN-PLAN." International Journal of Structural Stability and Dynamics 09, no. 02 (June 2009): 213–41. http://dx.doi.org/10.1142/s0219455409003004.
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The elastic response of curved beams subjected to moving vertical loads and dynamic loads is investigated. Incremental dynamic equilibrium equations are derived by using the principle of virtual work. Newmark's step-by-step procedure is adopted to discretise the dynamic equilibrium equations and obtain the time history response. Geometric nonlinearities due to large deflections and rotations are taken into account. A total Lagrangian finite element formulation is developed. The numerical models are compared with the existing analytical solutions and employed to show the effects of geometric nonlinearities as well as the initial curvature on the dynamic behaviour of curved I-beams. It is shown that the geometric nonlinearities are significant even for service loads. The nonlinear behaviour of a curved beam is substantially different from the nonlinear behaviour of a straight beam when the initial curvature is not small.
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LIU, Y. P., and J. N. REDDY. "A NONLOCAL CURVED BEAM MODEL BASED ON A MODIFIED COUPLE STRESS THEORY." International Journal of Structural Stability and Dynamics 11, no. 03 (June 2011): 495–512. http://dx.doi.org/10.1142/s0219455411004233.
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A nonlocal Timoshenko curved beam model is developed using a modified couple stress theory and Hamilton's principle. The model contains a material length scale parameter that can capture the size effect, unlike the classical Timoshenko beam theory. Both bending and axial deformations are considered, and the Poisson effect is incorporated in the model. The newly developed nonlocal model recovers the classical model when the material length scale parameter and Poisson's ratio are both taken to be zero and the straight beam model when the radius of curvature is set to infinity. In addition, the nonlocal Bernoulli–Euler curved beam model can be realized when the normal cross-section assumption is restated. To illustrate the new model, the static bending and free vibration problems of a simply supported curved beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko curved beam model. Also, the differences in both the deflection and rotation predicted by the current and classical Timoshenko model are very large when the beam thickness is small, but they diminish with the increase of the beam height. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the nonlocal model is higher than that by the classical model, and the difference between them is significantly large only for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed experimentally.
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Block, Joseph M., and Leon M. Keer. "Partial Plane Contact of an Elastic Curved Beam Pressed by a Flat Surface." Journal of Tribology 129, no. 1 (August 31, 2006): 60–64. http://dx.doi.org/10.1115/1.2401212.
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The normal contact of a frictionless, elastic curved beam indented by a flat, rigid surface is solved using a Michell–Fourier series expansion, which satisfies the mixed boundary value problem resulting from partial contact. When the contact region is small compared to the radius of curvature of the beam, semi-analytical solutions are obtained by exploiting dual series equation techniques. The relation between the level of loading and the extent of contact, as well as stress on the surface, are found for plane strain. The elasticity results extend Hertz line contact to finite thickness, curved beams. As the beam becomes thin, beam theory type behavior is recovered. The results may have application to finite-thickness wavy surfaces, cylindrical structures, or pressurized seals.
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Ecsedi, István, and Ákos József Lengyel. "An analytical solution for static problems of curved composite beams." Curved and Layered Structures 6, no. 1 (January 1, 2019): 105–16. http://dx.doi.org/10.1515/cls-2019-0009.
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AbstractAn analytical solution is presented for the determination of deformation of curved composite beams. Each cross-section is assumed to be symmetrical and the applied loads are acted in the plane of symmetry of curved beam. In-plane deformations are considered of composite curved beams. Assumed form of the displacement field assures the fulfillment of the classical Bernoulli-Euler beam theory. The curvature of beam is constant and the internal forces in a cross-section is replaced by an equivalent forcecouple system at the origin of the cylindrical coordinate system used. The internal forces are expressed in terms of two kinematical variables, which are the radial displacement and the rotation of the cross-sections. The determination of the analytical solutions of the considered static problems are based on the fundamental solutions. Linear combination of the fundamental solutions which are filling to the given loading and boundary conditions, gives the total solution. Closed form formulae are derived for the radial displacement, cross-sectional rotation, nomral and shear forces and bending moments. The circumferential and radial normal stresses and shear stresses are obtained by the integration of equilibrium equations. Examples illustrate the developed method.
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Avhad, Pravin V., and Atteshamuddin S. Sayyad. "On the deformation of laminated composite and sandwich curved beams." Curved and Layered Structures 9, no. 1 (October 18, 2021): 1–12. http://dx.doi.org/10.1515/cls-2022-0001.
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Abstract Plenty of research articles are available on the static deformation analysis of laminated straight beams using refined shear deformation theories. However, research on the deformation of laminated curved beams with simply supported boundary conditions is limited and needs more attention nowadays. With this objective, the present study deals with the static analysis of laminated composite and sandwich beams curved in elevation using a new quasi-3D polynomial type beam theory. The theory considers the effects of both transverse shear and normal strains, i.e. thickness stretching effects. In the present theory, axial displacement has expanded up to the fifth-order polynomial in terms of thickness coordinates to effectively account for the effects of curvature and deformations. The present theory satisfies the zero traction boundary condition on the top and bottom surfaces of the beam. Governing differential equations and associated boundary conditions are established by using the Principal of virtual work. Navier’s solution technique is used to obtain displacements and stresses for simply supported beams curved in elevation and subjected to uniformly distributed load. The present results can be benefited to the upcoming researchers.
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Youkhanna, Kanaan. "Onsite Load Bearing Capacity of Curved-up versus Straigt Beams." European Scientific Journal, ESJ 19, no. 24 (August 31, 2023): 15. http://dx.doi.org/10.19044/esj.2023.v19n24p15.
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Straight and curved-up beams are cast as part of the frame models. The models are made as single-span, double-span, and triple-span models. Experimental investigation is conducted to predict load capacity of curved-up beams compared to straight beams. Six models: single-span, double-span and triple-span are presented. Onsite, masonry blocks and bricks, steel pipes and gravel are used to load the beams uniformly. Load is applied gradually until failure mechanism. Enhancement to load capacity is observed in the range between 8.67% to 14.12%. The average enhancement ratio can be taken as 11% due to curved-up effect. Load capacity of interior curved-up beam/span will enhance by 120.3% and 131.3% for both straight and curved-up beams.
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Savino, Pierclaudio, Francesco Tondolo, Marco Gherlone, and Alexander Tessler. "Application of Inverse Finite Element Method to Shape Sensing of Curved Beams." Sensors 20, no. 24 (December 8, 2020): 7012. http://dx.doi.org/10.3390/s20247012.
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Curved beam, plate, and shell finite elements are commonly used in the finite element modeling of a wide range of civil and mechanical engineering structures. In civil engineering, curved elements are used to model tunnels, arch bridges, pipelines, and domes. Such structures provide a more efficient load transfer than their straight/flat counterparts due to the additional strength provided by their curved geometry. The load transfer is characterized by the bending, shear, and membrane actions. In this paper, a higher-order curved inverse beam element is developed for the inverse Finite Element Method (iFEM), which is aimed at reconstructing the deformed structural shapes based on real-time, in situ strain measurements. The proposed two-node inverse beam element is based on the quintic-degree polynomial shape functions that interpolate the kinematic variables. The element is C2 continuous and has rapid convergence characteristics. To assess the element predictive capabilities, several circular arch structures subjected to static loading are analyzed, under the assumption of linear elasticity and isotropic material behavior. Comparisons between direct FEM and iFEM results are presented. It is demonstrated that the present inverse beam finite element is both efficient and accurate, requiring only a few element subdivisions to reconstruct an accurate displacement field of shallow and deep curved beams.
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Changizi, Amin, Ion Stiharu, Bilal Outirba, and Patrick Hendrick. "Mathematical model of brush seals for gas turbine engines: A nonlinear analytical solution." Advances in Mechanical Engineering 13, no. 9 (September 2021): 168781402110433. http://dx.doi.org/10.1177/16878140211043396.
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Presented herein is a mathematical model employing differential equations formulation for brush seals used in gas turbine engines. These components are used to seal the bearing chamber from the environment and reduce the loss of lubricant in the atmosphere, ensuring a MTBR long enough to have required the change the seals only during the engine overhaul operation. The model assumes a single curved bristle loop in the form of a curved-bridge beam subjected to the influences of complex external loads (static and dynamic). Further, a model for clustered bristles is proposed. Specifically, the static forces acting on the curved-bridge beam include the weight of the oil capillary attached to the beam, the weight of the beam itself, the capillary force developed between the surfaces of the bristles in the brush and the temperature gradient. The dynamic forces include the leakage oil pressure and the rotation of the shaft. This complex loading induces a nonlinear large deflection on the curved-bridge beam. Also, the temperature gradient present on the bristles during the gas turbine engine operation generates a change in the geometry of the beam and in the magnitude of the forces acting on the bristles modeled as beams. In the present model, the weights are assumed as uniformly distributed forces on the surface of the beam while the capillary forces and the force generated by the rotating shaft are considered to be non-uniform. The equation expressing the curvature of the beam under general loading force is developed and one can choose the appropriate method of solving the generated differential equation after the expression of the general force is defined. Hence, the ordinary differential equation describing the nonlinear large deflection of the curved-bridge beam will be derived using general nonlinear elasticity theory.