With the third innovation in science and technology worldwide, China has also experienced thismarvelous progress. Concerning the longwall mining in China, the "masonry beam theory" (MBT) wasfirst proposed in the 1...With the third innovation in science and technology worldwide, China has also experienced thismarvelous progress. Concerning the longwall mining in China, the "masonry beam theory" (MBT) wasfirst proposed in the 1960s, illustrating that the transmission and equilibrium method of overburdenpressure using reserved coal pillar in mined-out areas can be realized. This forms the so-called "121mining method", which lays a solid foundation for development of mining science and technology inChina. The "transfer rock beam theory" (TRBT) proposed in the 1980s gives a further understanding forthe transmission path of stope overburden pressure and pressure distribution in high-stress areas. In thisregard, the advanced 121 mining method was proposed with smaller coal pillar for excavation design,making significant contributions to improvement of the coal recovery rate in that era. In the 21st century,the traditional mining technologies faced great challenges and, under the theoretical developmentspioneered by Profs. Minggao Qian and Zhenqi Song, the "cutting cantilever beam theory" (CCBT) wasproposed in 2008. After that the 110 mining method is formulated subsequently, namely one stope face,after the first mining cycle, needs one advanced gateway excavation, while the other one is automaticallyformed during the last mining cycle without coal pillars left in the mining area. This method can beimplemented using the CCBT by incorporating the key technologies, including the directional presplittingroof cutting, constant resistance and large deformation (CRLD) bolt/anchor supporting systemwith negative Poisson's ratio (NPR) effect material, and remote real-time monitoring technology. TheCCBT and 110 mining method will provide the theoretical and technical basis for the development ofmining industry in China.展开更多
Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equati...Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.展开更多
Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a...Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.展开更多
A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain g...A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.展开更多
This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and tw...This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.展开更多
To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy's higher order beam theory. The governing differentia...To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy's higher order beam theory. The governing differential equations for free vibration and buckling are formulated using the Hamilton's principle, the natural frequencies and axial forces are thus analytically obtained by Laplace transform technique. The analytical results are verified through the comparison with those of several other models common in use; and the presented model is found to be a finer one than the Reddy's. A parametric study is also performed to investigate the effects of geometry and material parameters.展开更多
In Order to study the frictional contact problems of the elastoplastic beam theory,an extended two-dimensional beam model is established, and a second order nonlinear equilibrium problem with both internal and exter...In Order to study the frictional contact problems of the elastoplastic beam theory,an extended two-dimensional beam model is established, and a second order nonlinear equilibrium problem with both internal and external complementarity conditions is proposed. The external complementarity condition provides the free boundary condition. while the internal complemententarity condition gives the interface of the elastic and plastic regions. We prove that this bicomplementarity problem is equivalent to a nonlinear variational inequality The dual variational inequality is also developed.It is shown that the dual variational inequality is much easier than the primalvariational problem. Application to limit analysis is illustrated.展开更多
The exact relationship between the bending solutions of functionally graded material (FGM) beams based on the Levinson beam theory and those of the correspond- ing homogenous beams based on the classical beam theory...The exact relationship between the bending solutions of functionally graded material (FGM) beams based on the Levinson beam theory and those of the correspond- ing homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The de- flection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams (FGMLBs) are given analytically in terms of the deflection of the reference ho- mogenous Euler-Bernoulli beams (HEBBs) with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily deter- mined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be eas- ily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.展开更多
In the present paper, the isogeometric analysis(IGA) of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature...In the present paper, the isogeometric analysis(IGA) of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables(displacement and rotation) in a finite element analysis are considered to be the same as the non-uniform rational basis spline(NURBS) basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.展开更多
By incorporating the strain gradient elasticity into the classical Bernoulli-Euler beam and Timoshenko beam models, the size-dependent characteristics of wave propaga- tion in micro/nanobeams is studied. The formulati...By incorporating the strain gradient elasticity into the classical Bernoulli-Euler beam and Timoshenko beam models, the size-dependent characteristics of wave propaga- tion in micro/nanobeams is studied. The formulations of dis- persion relation are explicitly derived for both strain gradi- ent beam models, and presented for different material length scale parameters (MLSPs). For both phenomenological size- dependent beam models, the angular frequency, phase veloc- ity and group velocity increase with increasing wave num- ber. However, the velocity ratios approach different values for different beam models, indicating an interesting behavior of the asymptotic velocity ratio. The present theory is also compared with the nonlocal continuum beam models.展开更多
The bending responses of functionally graded (FG) nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to an...The bending responses of functionally graded (FG) nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index, and surface stress on dimensionless deflection are discussed in detail.展开更多
The bending and free vibrational behaviors of functionally graded(FG)cylindrical beams with radially and axially varying material inhomogeneities are investigated.Based on a high-order cylindrical beam model,where the...The bending and free vibrational behaviors of functionally graded(FG)cylindrical beams with radially and axially varying material inhomogeneities are investigated.Based on a high-order cylindrical beam model,where the shear deformation and rotary inertia are both considered,the two coupled governing differential motion equations for the deflection and rotation are established.The analytical bending solutions for various boundary conditions are derived.In the vibrational analysis of FG cylindrical beams,the two governing equations are firstly changed to a single equation by means of an auxiliary function,and then the vibration mode is expanded into shifted Chebyshev polynomials.Numerical examples are given to investigate the effects of the material gradient indices on the deflections,the stress distributions,and the eigenfrequencies of the cylindrical beams,respectively.By comparing the obtained numerical results with those obtained by the three-dimensional(3D)elasticity theory and the Timoshenko beam theory,the effectiveness of the present approach is verified.展开更多
In this study, a model for dynamic instability of embedded single-walled car- bon nanotubes (SWCNTs) is presented. SWCNTs are modeled by the sinusoidal shear deformation beam theory (SSDBT). The modified couple st...In this study, a model for dynamic instability of embedded single-walled car- bon nanotubes (SWCNTs) is presented. SWCNTs are modeled by the sinusoidal shear deformation beam theory (SSDBT). The modified couple stress theory (MCST) is con- sidered in order to capture the size effects. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear, and damping loads. The motion equations are derived based on Hamilton's principle. The differential quadrature method (DQM) in conjunction with the Bolotin method is used in order to calculate the dynamic instability region (DIR) of SWCNTs. The effects of differ- ent parameters, such as nonlocal parameter, visco-Pasternak foundation, mode numbers, and geometrical parameters, are shown on the dynamic instability of SWCNTs. The re- sults depict that increasing the nonlocal parameter shifts the DIR to right. The results presented in this paper would be helpful in design and manufacturing of nano-electromechanical system (NEMS) and micro-electro-mechanical system (MEMS).展开更多
This paper extends Le van's work to the case of nonlinear problem and the complicated configuration. The wrinkling stress distribution and the pressure effects are also included in our analysis. Pseudo-beam method is...This paper extends Le van's work to the case of nonlinear problem and the complicated configuration. The wrinkling stress distribution and the pressure effects are also included in our analysis. Pseudo-beam method is presented based on the inflatable beam theory to model the inflatable structures as a set of inflatable beam elements with a prestressed state. In this method, the discretized nonlinear equations are given based upon the virtual work principle with a 3-node Timoshenko's beam model. Finite element simulation is performed by using a 3-node BEAM189 element incorporating ANSYS nonlinear program. The pressure effect is equivalent included in our method by modifying beam element cross-section parameters related to pressure. A benchmark example, the bending case of an inflatable cantilever beam, is performed to verify the accuracy of our proposed method. The comparisons reveal that the numerical results obtained with our method are close to open published analytical and membrane finite element results. The method is then used to evaluate the whole buckling and the loadcarrying characteristics of an inflatable support frame subjected to a compression force. The wrinkling stress and region characteristics are also shown in the end. This method gives better convergence characteristics, and requires much less computation time. It is very effective to deal with the whole load-carrying ability analytical problems for large scale inflatable structures with complex configuration.展开更多
Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is base...Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is based on the nonlocal Euler–Bernoulli beam theory. The eigen value equation is developed for the buckling and vibration analyses. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient,and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes.展开更多
Advancements in manufacturing technology,including the rapid development of additive manufacturing(AM),allow the fabrication of complex functionally graded material(FGM)sectioned beams.Portions of these beams may be m...Advancements in manufacturing technology,including the rapid development of additive manufacturing(AM),allow the fabrication of complex functionally graded material(FGM)sectioned beams.Portions of these beams may be made from different materials with possibly different gradients of material properties.The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional(1D)finite element analysis.For this purpose,a sample beam is divided into discrete elements,and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory.Then,Hamilton’s principle is used to derive the equations of motion for the beam.The effects of material properties and dimensions of FGM sections on the beam’s natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model(TM).The presented model is validated by comparison with three-dimensional(3D)finite element simulations of the first three mode shapes of the beam.展开更多
This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of ...This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange's equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.展开更多
The importance of the geometrical effect in practical design has been evaluated, showing that an overestimation of the actual member rotation is very likely if the available rotation capacity is based on the evaluatio...The importance of the geometrical effect in practical design has been evaluated, showing that an overestimation of the actual member rotation is very likely if the available rotation capacity is based on the evaluation of the behavior of the reference members within a limited size range. The increase of ductility with decreasing member size has been interpreted in fracture mechanics of reinforced concrete. In fracture mechanics it’s seen that beams with higher dimensions are brittle, while those with small dimensions are ductile, so it’s important here to clarify if the same material and design concepts can be applied for reinforced concrete beams with different scales. Three point bending test was executed on 20 reinforced concrete beams varying scale and slenderness ratio (where steel ratio being kept constant). The experimental results obtained varying beam slenderness and beam depth will be used to analyze the structural response for a practical construction, taking in consideration the size effect, these beams are normally designed in such a way that the distribution of their internal forces over the transversal section has been calculated as per elastic beam theory, while the beam dimension will be designed as per the ultimate limit state to obtain a ductile response of the reinforced concrete beams which is necessary to guarantee the structural safety [1].展开更多
This paper focuses on the size-dependently mechanical behaviors of a micro-beam under forced vibration.Governing equations of a micro-beam under forced vibration are established by using the modified couple stress the...This paper focuses on the size-dependently mechanical behaviors of a micro-beam under forced vibration.Governing equations of a micro-beam under forced vibration are established by using the modified couple stress theory,Bernoulli-Euler beam theory and D’Alembert’s principle together.A simply supported micro-beam under forced vibration is solved according to the established governing equations and the method of separation of variables.The dimensionless deflection,amplitude mode and period mode are defined to investigate the size-dependently mechanical behaviors of a micro-beam under forced vibration.Results show that the performance of a micro-beams under forced vibration is distinctly size-dependent when the ratio of micro-beam height to material length-scale parameter is small enough.Both frequency ratio and loading location are the important factors that determine the sizedependent performance of a micro-beams under forced vibration.展开更多
We present the application of differential quadrature(DQ) method for the buckling analysis of nanobeams with exponentially varying stiffness based on four different beam theories of Euler-Bernoulli, Timoshenko, Redd...We present the application of differential quadrature(DQ) method for the buckling analysis of nanobeams with exponentially varying stiffness based on four different beam theories of Euler-Bernoulli, Timoshenko, Reddy, and Levison.The formulation is based on the nonlocal elasticity theory of Eringen. New results are presented for the guided and simply supported guided boundary conditions. Numerical results are obtained to investigate the effects of the nonlocal parameter,length-to-height ratio, boundary condition, and nonuniform parameter on the critical buckling load parameter. It is observed that the critical buckling load decreases with increase in the nonlocal parameter while the critical buckling load parameter increases with increase in the length-to-height ratio.展开更多
基金supported by the National Natural Science Foundation of China (No. 51404278)the State Key Program of National Natural Science Foundation of China (No. 51134005)
文摘With the third innovation in science and technology worldwide, China has also experienced thismarvelous progress. Concerning the longwall mining in China, the "masonry beam theory" (MBT) wasfirst proposed in the 1960s, illustrating that the transmission and equilibrium method of overburdenpressure using reserved coal pillar in mined-out areas can be realized. This forms the so-called "121mining method", which lays a solid foundation for development of mining science and technology inChina. The "transfer rock beam theory" (TRBT) proposed in the 1980s gives a further understanding forthe transmission path of stope overburden pressure and pressure distribution in high-stress areas. In thisregard, the advanced 121 mining method was proposed with smaller coal pillar for excavation design,making significant contributions to improvement of the coal recovery rate in that era. In the 21st century,the traditional mining technologies faced great challenges and, under the theoretical developmentspioneered by Profs. Minggao Qian and Zhenqi Song, the "cutting cantilever beam theory" (CCBT) wasproposed in 2008. After that the 110 mining method is formulated subsequently, namely one stope face,after the first mining cycle, needs one advanced gateway excavation, while the other one is automaticallyformed during the last mining cycle without coal pillars left in the mining area. This method can beimplemented using the CCBT by incorporating the key technologies, including the directional presplittingroof cutting, constant resistance and large deformation (CRLD) bolt/anchor supporting systemwith negative Poisson's ratio (NPR) effect material, and remote real-time monitoring technology. TheCCBT and 110 mining method will provide the theoretical and technical basis for the development ofmining industry in China.
基金supported by the National Natural Science Foundation of China(No.11272278)
文摘Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.
基金Project supported by the National Basic Research Program of China (No. 2011CB610300)the National Natural Science Foundation of China (Nos. 10972182, 11172239, and 10902089)+3 种基金the 111 Project of China (No. B07050)the Ph. D. Programs Foundation of Ministry of Education of China (No. 20106102110019)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (No. GZ0802)the Doctorate Foundation of Northwestern Polytechnical University (No. CX201224)
文摘Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube (SWCNT) subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.
基金Project supported by the National Natural Science Foundation of China (No. 12002086)the Fundamental Research Funds for the Central Universities of China (No. 2242022R40040)。
文摘A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.
文摘This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.
基金Project supported by the National High Technology Research and Development Program of China(No.2009AA032303-2)
文摘To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy's higher order beam theory. The governing differential equations for free vibration and buckling are formulated using the Hamilton's principle, the natural frequencies and axial forces are thus analytically obtained by Laplace transform technique. The analytical results are verified through the comparison with those of several other models common in use; and the presented model is found to be a finer one than the Reddy's. A parametric study is also performed to investigate the effects of geometry and material parameters.
文摘In Order to study the frictional contact problems of the elastoplastic beam theory,an extended two-dimensional beam model is established, and a second order nonlinear equilibrium problem with both internal and external complementarity conditions is proposed. The external complementarity condition provides the free boundary condition. while the internal complemententarity condition gives the interface of the elastic and plastic regions. We prove that this bicomplementarity problem is equivalent to a nonlinear variational inequality The dual variational inequality is also developed.It is shown that the dual variational inequality is much easier than the primalvariational problem. Application to limit analysis is illustrated.
基金supported by the National Natural Science Foundation of China(No.11272278)
文摘The exact relationship between the bending solutions of functionally graded material (FGM) beams based on the Levinson beam theory and those of the correspond- ing homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The de- flection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams (FGMLBs) are given analytically in terms of the deflection of the reference ho- mogenous Euler-Bernoulli beams (HEBBs) with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily deter- mined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be eas- ily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.
文摘In the present paper, the isogeometric analysis(IGA) of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables(displacement and rotation) in a finite element analysis are considered to be the same as the non-uniform rational basis spline(NURBS) basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.
基金supported by the National Natural Science Foundation of China(11202117,11272186,11172231 and 50928601)the Postdoctoral Science Foundation of China(2012M521326)+3 种基金the Natural Science Fund of Shandong Province(ZR2012AM014 and BS2012ZZ006)Independent Innovation Fund of Shandong University(2011GN055)National Science Foundation(CMMI-0643726),DARPA(W91CRB-11-C-0112)Changjiang Scholar Program from Ministry of Education of China
文摘By incorporating the strain gradient elasticity into the classical Bernoulli-Euler beam and Timoshenko beam models, the size-dependent characteristics of wave propaga- tion in micro/nanobeams is studied. The formulations of dis- persion relation are explicitly derived for both strain gradi- ent beam models, and presented for different material length scale parameters (MLSPs). For both phenomenological size- dependent beam models, the angular frequency, phase veloc- ity and group velocity increase with increasing wave num- ber. However, the velocity ratios approach different values for different beam models, indicating an interesting behavior of the asymptotic velocity ratio. The present theory is also compared with the nonlocal continuum beam models.
基金supported by the National Natural Science Foundation of China(11302055)Heilongjiang Post-doctoral Scientific Research Start-up Funding(LBH-Q14046)
文摘The bending responses of functionally graded (FG) nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index, and surface stress on dimensionless deflection are discussed in detail.
基金Project supported by the Natural Science Foundation of Guangdong Province of China(No.2018A030313258)。
文摘The bending and free vibrational behaviors of functionally graded(FG)cylindrical beams with radially and axially varying material inhomogeneities are investigated.Based on a high-order cylindrical beam model,where the shear deformation and rotary inertia are both considered,the two coupled governing differential motion equations for the deflection and rotation are established.The analytical bending solutions for various boundary conditions are derived.In the vibrational analysis of FG cylindrical beams,the two governing equations are firstly changed to a single equation by means of an auxiliary function,and then the vibration mode is expanded into shifted Chebyshev polynomials.Numerical examples are given to investigate the effects of the material gradient indices on the deflections,the stress distributions,and the eigenfrequencies of the cylindrical beams,respectively.By comparing the obtained numerical results with those obtained by the three-dimensional(3D)elasticity theory and the Timoshenko beam theory,the effectiveness of the present approach is verified.
文摘In this study, a model for dynamic instability of embedded single-walled car- bon nanotubes (SWCNTs) is presented. SWCNTs are modeled by the sinusoidal shear deformation beam theory (SSDBT). The modified couple stress theory (MCST) is con- sidered in order to capture the size effects. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear, and damping loads. The motion equations are derived based on Hamilton's principle. The differential quadrature method (DQM) in conjunction with the Bolotin method is used in order to calculate the dynamic instability region (DIR) of SWCNTs. The effects of differ- ent parameters, such as nonlocal parameter, visco-Pasternak foundation, mode numbers, and geometrical parameters, are shown on the dynamic instability of SWCNTs. The re- sults depict that increasing the nonlocal parameter shifts the DIR to right. The results presented in this paper would be helpful in design and manufacturing of nano-electromechanical system (NEMS) and micro-electro-mechanical system (MEMS).
基金supported by the Specialized Fund for the Doctoral Program of Higher Education of China (200802131046)China Postdoctoral Science Foundation Funded Major Project (200801290)+1 种基金Development Program of Outstanding Young Teachers in Harbin Institute of Technology (HITQNJS.2008.004)Specialized Fund for Innovation Talents of Science and Technology in Harbin (2008RFQXG057).
文摘This paper extends Le van's work to the case of nonlinear problem and the complicated configuration. The wrinkling stress distribution and the pressure effects are also included in our analysis. Pseudo-beam method is presented based on the inflatable beam theory to model the inflatable structures as a set of inflatable beam elements with a prestressed state. In this method, the discretized nonlinear equations are given based upon the virtual work principle with a 3-node Timoshenko's beam model. Finite element simulation is performed by using a 3-node BEAM189 element incorporating ANSYS nonlinear program. The pressure effect is equivalent included in our method by modifying beam element cross-section parameters related to pressure. A benchmark example, the bending case of an inflatable cantilever beam, is performed to verify the accuracy of our proposed method. The comparisons reveal that the numerical results obtained with our method are close to open published analytical and membrane finite element results. The method is then used to evaluate the whole buckling and the loadcarrying characteristics of an inflatable support frame subjected to a compression force. The wrinkling stress and region characteristics are also shown in the end. This method gives better convergence characteristics, and requires much less computation time. It is very effective to deal with the whole load-carrying ability analytical problems for large scale inflatable structures with complex configuration.
文摘Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is based on the nonlocal Euler–Bernoulli beam theory. The eigen value equation is developed for the buckling and vibration analyses. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient,and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes.
基金Project supported by Khalifa University of Science and Technology(No.CIRA 2019-024)。
文摘Advancements in manufacturing technology,including the rapid development of additive manufacturing(AM),allow the fabrication of complex functionally graded material(FGM)sectioned beams.Portions of these beams may be made from different materials with possibly different gradients of material properties.The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional(1D)finite element analysis.For this purpose,a sample beam is divided into discrete elements,and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory.Then,Hamilton’s principle is used to derive the equations of motion for the beam.The effects of material properties and dimensions of FGM sections on the beam’s natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model(TM).The presented model is validated by comparison with three-dimensional(3D)finite element simulations of the first three mode shapes of the beam.
文摘This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange's equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.
文摘The importance of the geometrical effect in practical design has been evaluated, showing that an overestimation of the actual member rotation is very likely if the available rotation capacity is based on the evaluation of the behavior of the reference members within a limited size range. The increase of ductility with decreasing member size has been interpreted in fracture mechanics of reinforced concrete. In fracture mechanics it’s seen that beams with higher dimensions are brittle, while those with small dimensions are ductile, so it’s important here to clarify if the same material and design concepts can be applied for reinforced concrete beams with different scales. Three point bending test was executed on 20 reinforced concrete beams varying scale and slenderness ratio (where steel ratio being kept constant). The experimental results obtained varying beam slenderness and beam depth will be used to analyze the structural response for a practical construction, taking in consideration the size effect, these beams are normally designed in such a way that the distribution of their internal forces over the transversal section has been calculated as per elastic beam theory, while the beam dimension will be designed as per the ultimate limit state to obtain a ductile response of the reinforced concrete beams which is necessary to guarantee the structural safety [1].
基金The authors of this paper acknowledge the supports from the National Key Research and Development Program of China(Grant No.2017YFC0307604)the Talent Foundation of China University of Petroleum(Grant No.Y1215042).
文摘This paper focuses on the size-dependently mechanical behaviors of a micro-beam under forced vibration.Governing equations of a micro-beam under forced vibration are established by using the modified couple stress theory,Bernoulli-Euler beam theory and D’Alembert’s principle together.A simply supported micro-beam under forced vibration is solved according to the established governing equations and the method of separation of variables.The dimensionless deflection,amplitude mode and period mode are defined to investigate the size-dependently mechanical behaviors of a micro-beam under forced vibration.Results show that the performance of a micro-beams under forced vibration is distinctly size-dependent when the ratio of micro-beam height to material length-scale parameter is small enough.Both frequency ratio and loading location are the important factors that determine the sizedependent performance of a micro-beams under forced vibration.
文摘We present the application of differential quadrature(DQ) method for the buckling analysis of nanobeams with exponentially varying stiffness based on four different beam theories of Euler-Bernoulli, Timoshenko, Reddy, and Levison.The formulation is based on the nonlocal elasticity theory of Eringen. New results are presented for the guided and simply supported guided boundary conditions. Numerical results are obtained to investigate the effects of the nonlocal parameter,length-to-height ratio, boundary condition, and nonuniform parameter on the critical buckling load parameter. It is observed that the critical buckling load decreases with increase in the nonlocal parameter while the critical buckling load parameter increases with increase in the length-to-height ratio.