This study proposes an effective method to enhance the accuracy of the Differential Quadrature Method(DQM)for calculating the dynamic characteristics of functionally graded beams by improving the form of discrete node...This study proposes an effective method to enhance the accuracy of the Differential Quadrature Method(DQM)for calculating the dynamic characteristics of functionally graded beams by improving the form of discrete node distribution.Firstly,based on the first-order shear deformation theory,the governing equation of free vibration of a functionally graded beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam axial displacement,transverse displacement,and cross-sectional rotation angle by considering the effects of shear deformation and rotational inertia of the beam cross-section.Then,ignoring the shear deformation of the beam section and only considering the effect of the rotational inertia of the section,the governing equation of the beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam transverse displacement.Based on the differential quadrature method theory,the eigenvalue problem of ordinary differential equations is transformed into the eigenvalue problem of standard generalized algebraic equations.Finally,the first several natural frequencies of the beam can be calculated.The feasibility and accuracy of the improved DQM are verified using the finite element method(FEM)and combined with the results of relevant literature.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
Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transve...Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely nonuniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.展开更多
Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexu...Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam＇s slenderness ratio, the crack＇s depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam＇s deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam＇s slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.展开更多
The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by con...The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.展开更多
The thermal vibration of functionally graded(FG)porous nanocomposite beams reinforced by graphene platelets(GPLs)is studied.The beams are exposed to the thermal gradient with a multilayer structure.The temperature var...The thermal vibration of functionally graded(FG)porous nanocomposite beams reinforced by graphene platelets(GPLs)is studied.The beams are exposed to the thermal gradient with a multilayer structure.The temperature varies linearly across the thickness direction.Three different types of dispersion patterns of GPLs as well as porosity distributions are presented.The material properties vary along the thickness direction.By using the mechanical parameters of closed-cell cellular solid,the variation of Poisson’s ratio and the relation between the porosity coefficient and the mass density under the Gaussian random field(GRF)model are obtained.By using the Halpin-Tsai micromechanics model,the elastic modulus of the nanocomposite is achieved.The equations of motion based on the Timoshenko beam theory are obtained by using Hamilton’s principle.These equations are discretized and solved by using the generalized differential quadrature method(GDQM)to obtain the fundamental frequencies.The effects of the weight fraction,the dispersion model,the geometry,and the size of GPLs,as well as the porosity distribution,the porosity coefficient,the boundary condition,the metal matrix,the slenderness ratio,and the thermal gradient are presented.展开更多
Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-kimwn that the system is exponentially stable if the kernel in the memory term is sub- exponential. That is, if the product of the ker...Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-kimwn that the system is exponentially stable if the kernel in the memory term is sub- exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions： What if the kernel is tested against a different function （say Gamma） other than the exponential function？ Would there still be stability？ In the affirmative, what kind of decay rate we get？ It is proved that for a non- decreasing function ＂Gamma＂ whose ＂logarithmic derivative＂ is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.展开更多
Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling w...Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.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
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
基金Anhui Provincial Natural Science Foundation(2308085QD124)Anhui Province University Natural Science Research Project(GrantNo.2023AH050918)The University Outstanding Youth Talent Support Program of Anhui Province.
文摘This study proposes an effective method to enhance the accuracy of the Differential Quadrature Method(DQM)for calculating the dynamic characteristics of functionally graded beams by improving the form of discrete node distribution.Firstly,based on the first-order shear deformation theory,the governing equation of free vibration of a functionally graded beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam axial displacement,transverse displacement,and cross-sectional rotation angle by considering the effects of shear deformation and rotational inertia of the beam cross-section.Then,ignoring the shear deformation of the beam section and only considering the effect of the rotational inertia of the section,the governing equation of the beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam transverse displacement.Based on the differential quadrature method theory,the eigenvalue problem of ordinary differential equations is transformed into the eigenvalue problem of standard generalized algebraic equations.Finally,the first several natural frequencies of the beam can be calculated.The feasibility and accuracy of the improved DQM are verified using the finite element method(FEM)and combined with the results of relevant literature.
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 National Natural Science Foundation of China (No.10472039)
文摘Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely nonuniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.
文摘Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam＇s slenderness ratio, the crack＇s depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam＇s deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam＇s slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.
基金Project supported by the National Natural Science Foundation of China(No.11272278)
文摘The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.
文摘The thermal vibration of functionally graded(FG)porous nanocomposite beams reinforced by graphene platelets(GPLs)is studied.The beams are exposed to the thermal gradient with a multilayer structure.The temperature varies linearly across the thickness direction.Three different types of dispersion patterns of GPLs as well as porosity distributions are presented.The material properties vary along the thickness direction.By using the mechanical parameters of closed-cell cellular solid,the variation of Poisson’s ratio and the relation between the porosity coefficient and the mass density under the Gaussian random field(GRF)model are obtained.By using the Halpin-Tsai micromechanics model,the elastic modulus of the nanocomposite is achieved.The equations of motion based on the Timoshenko beam theory are obtained by using Hamilton’s principle.These equations are discretized and solved by using the generalized differential quadrature method(GDQM)to obtain the fundamental frequencies.The effects of the weight fraction,the dispersion model,the geometry,and the size of GPLs,as well as the porosity distribution,the porosity coefficient,the boundary condition,the metal matrix,the slenderness ratio,and the thermal gradient are presented.
基金the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through project No. IN111034
文摘Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-kimwn that the system is exponentially stable if the kernel in the memory term is sub- exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions： What if the kernel is tested against a different function （say Gamma） other than the exponential function？ Would there still be stability？ In the affirmative, what kind of decay rate we get？ It is proved that for a non- decreasing function ＂Gamma＂ whose ＂logarithmic derivative＂ is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.
基金Project supported by the National Natural Science Foundation of China (No. 10772129)
文摘Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.
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.
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.
