Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled diff...Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.展开更多
The present study deals with free vibration analysis of variable thickness viscoelastic circular plates made of heterogeneous materials and resting on two-parameter elastic foundations in addition to their edge condit...The present study deals with free vibration analysis of variable thickness viscoelastic circular plates made of heterogeneous materials and resting on two-parameter elastic foundations in addition to their edge conditions. It is assumed that the viscoelastic material properties vary in the transverse and radial directions simultaneously. The complex modulus approach is employed in conjunction with the elastic-viscoelastic correspondence principle to obtain the solution. The governing equations are solved by means of a power series solution. Finally, a sensitivity analysis including evaluation of effects of various edge conditions, thickness variations, coefficients of the elastic foundation, and material loss factor and heterogeneity on the natural frequencies and modal loss factors is accomplished.展开更多
In recent years,magneto-electro-elastic(MEE)cylindrical shells with step-wise thicknesses have shown significant potential in the field of vibration energy harvesting.To aid the design of such energy harvesting device...In recent years,magneto-electro-elastic(MEE)cylindrical shells with step-wise thicknesses have shown significant potential in the field of vibration energy harvesting.To aid the design of such energy harvesting devices,an accurate free vibration analysis of embedded MEE cylindrical shells with step-wise thicknesses is performed within the framework of symplectic mechanics.By using the Legendre transformation,a new known vector is defined to transform the higher-order partial differential governing equations into a set of lower-order ordinary differential equations.Therefore,the original vibration analysis is regarded as an eigen problem in the symplectic space,and analytical solutions can be represented by the symplectic series.In numerical examples,the new analytical solutions are compared with the existing results,and good agreement is observed.Furthermore,the effects of critical design parameters on free vibration characteristics are thoroughly investigated.All numerical results can serve as benchmarks for the development of other approximate or numerical methods.展开更多
In deep underground mining,achieving stable support for roadways along with long service life is critical and the complex geological environment at such depths frequently presents a major challenge.Owing to the coupli...In deep underground mining,achieving stable support for roadways along with long service life is critical and the complex geological environment at such depths frequently presents a major challenge.Owing to the coupling action of multiple factors such as deep high stress,adjacent faults,cross-layer design,weak lithology,broken surrounding rock,variable cross-sections,wide sections up to 9.9 m,and clusters of nearby chambers,there was severe deformation and breakdown in the No.10 intersection of the roadway of large-scale variable cross-section at the−760 m level in a coal mine.As there are insufcient examples in engineering methods pertaining to the geological environment described above,the numerical calculation model was oversimplifed and support theory underdeveloped;therefore,it is imperative to develop an efective support system for the stability and sustenance of deep roadways.In this study,a quantitative analysis of the geological environment of the roadway through feld observations,borehole-scoping,and ground stress testing is carried out to establish the FLAC 3D variable cross-section crossing roadway model.This model is combined with the strain softening constitutive(surrounding rock)and Mohr–Coulomb constitutive(other deep rock formations)models to construct a compression arch mechanical model for deep soft rock,based on the quadratic parabolic Mohr criterion.An integrated control technology of bolting and grouting that is mainly composed of a high-strength hollow grouting cable bolt equipped with modifed cement grouting materials and a high-elongation cable bolt is developed by analyzing the strengthening properties of the surrounding rock before and after bolting,based on the Heok-Brown criterion.As a result of on-site practice,the following conclusions are drawn:(1)The plastic zone of the roof of the cross roadway is approximately 6 m deep in this environment,the tectonic stress is nearly 30 MPa,and the surrounding rock is severely fractured.(2)The deformation of the roadway progressively increases from small to large cross-sections,almost doubling at the largest cross-section.The plastic zone is concentrated at the top plate and shoulder and decreases progressively from the two sides to the bottom corner.The range of stress concentration at the sides of the intersection roadway close to the passageway is wider and higher.(3)The 7 m-thick reinforced compression arch constructed under the strengthening support scheme has a bearing capacity enhanced by 1.8 to 2.3 times and increase in thickness of the bearing structure by 1.76 times as compared to the original scheme.(4)The increase in the mechanical parameters c andφof the surrounding rock after anchoring causes a signifcant increase inσt;the pulling force of the cable bolt beneath the new grouting material is more than twice that of ordinary cement grout,and according to the test,the supporting stress feld shows that the 7.24 m surrounding rock is compacted and strengthened in addition to providing a strong foundation for the bolt(cable).On-site monitoring shows that the 60-days convergence is less than 30 mm,indicating that the stability control of the roadway is successful.展开更多
In this paper two-dimensional differential quadrature method has been used to analyze thick Functionally Graded(FG)rotating disks with non-uniform boundary conditions and variable thickness.Material properties vary co...In this paper two-dimensional differential quadrature method has been used to analyze thick Functionally Graded(FG)rotating disks with non-uniform boundary conditions and variable thickness.Material properties vary continuously along both radial and axial directions by a power law pattern.Three-dimensional solid mechanics theory is employed to formulate the axisymmetric problem as a second order system of partial differential equations.The non-uniform boundary conditions are exerted directly into the governing equations to reach the eigenvalue system of equations.Four different disk profile shapes are considered and discussed.The effect of the power law exponent is also investigated and results show that by the use of material which functionally varied along the radial and especially axial directions the stresses and strains can be controlled so the capability of the disk is increased.Comparison with other available approaches in the literature shows a good agreement here in terms of computational time,robustness and accuracy of the present method.Moreover,novel applications are shown to provide results for further studies on the same topics.展开更多
This article presents the investigation of nonlinear vibration analysis of tapered porous functionally graded skew(TPFGS)plate considering the effects of geometrical non-uniformities to optimize the thickness in the s...This article presents the investigation of nonlinear vibration analysis of tapered porous functionally graded skew(TPFGS)plate considering the effects of geometrical non-uniformities to optimize the thickness in the structural design.The TPFGS plate is analyzed considering linearly,bi-linearly,and exponentially varying thicknesses.The plate’s effective material properties are tailor-made using a modified power-law distribution in which gradation varies along the thickness direction of the TPFGS plate.Incorporating the non-linear finite element formulation to develop the kinematic equation’s displacement model for the TPFGS plate is based on the first-order shear deformation theory(FSDT)in conjunction with von Karman’s nonlinearity.The nonlinear governing equations are established by Hamilton’s principle.The direct iterative method is adopted to solve the nonlinear mathematical relations to obtain the nonlinear frequencies.The influence of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the TPFGS plate for different skew angles and variable thicknesses are studied for various geometrical parameters.The influence of taper ratio,variable thickness,skewness,porosity distributions,gradation,and boundary conditions on the plate’s nonlinear vibration is demonstrated.The nonlinear frequency analysis reveals that the geometrical nonuniformities and porosities significantly influence the porous functionally graded plates with varying thickness than the uniform thickness.Besides,exponentially and linearly variable thicknesses can be considered for the thickness optimizations of TPFGS plates in the structural design.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11402133,11620162,11321202,and 11532001)
文摘Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.
文摘The present study deals with free vibration analysis of variable thickness viscoelastic circular plates made of heterogeneous materials and resting on two-parameter elastic foundations in addition to their edge conditions. It is assumed that the viscoelastic material properties vary in the transverse and radial directions simultaneously. The complex modulus approach is employed in conjunction with the elastic-viscoelastic correspondence principle to obtain the solution. The governing equations are solved by means of a power series solution. Finally, a sensitivity analysis including evaluation of effects of various edge conditions, thickness variations, coefficients of the elastic foundation, and material loss factor and heterogeneity on the natural frequencies and modal loss factors is accomplished.
基金Project supported by the Science and Technology Plan Joint Program of Liaoning Province of China(Natural Science Foundation-Doctoral Research Launch Project)(No.2024-BSLH-027)the Fundamental Research Funds for Undergraduate Universities of Liaoning Province of China(No.LJBKY2024033)+1 种基金the National Natural Science Foundation of China(No.12472064)the Natural Science Foundation of Liaoning Province of China(No.2023-MS-118)。
文摘In recent years,magneto-electro-elastic(MEE)cylindrical shells with step-wise thicknesses have shown significant potential in the field of vibration energy harvesting.To aid the design of such energy harvesting devices,an accurate free vibration analysis of embedded MEE cylindrical shells with step-wise thicknesses is performed within the framework of symplectic mechanics.By using the Legendre transformation,a new known vector is defined to transform the higher-order partial differential governing equations into a set of lower-order ordinary differential equations.Therefore,the original vibration analysis is regarded as an eigen problem in the symplectic space,and analytical solutions can be represented by the symplectic series.In numerical examples,the new analytical solutions are compared with the existing results,and good agreement is observed.Furthermore,the effects of critical design parameters on free vibration characteristics are thoroughly investigated.All numerical results can serve as benchmarks for the development of other approximate or numerical methods.
基金supported by the National Natural Science Foundation of China(Grant Nos.52074296,52004286)the China Postdoctoral Science Foundation(Grant Nos.2020T130701,2019M650895).
文摘In deep underground mining,achieving stable support for roadways along with long service life is critical and the complex geological environment at such depths frequently presents a major challenge.Owing to the coupling action of multiple factors such as deep high stress,adjacent faults,cross-layer design,weak lithology,broken surrounding rock,variable cross-sections,wide sections up to 9.9 m,and clusters of nearby chambers,there was severe deformation and breakdown in the No.10 intersection of the roadway of large-scale variable cross-section at the−760 m level in a coal mine.As there are insufcient examples in engineering methods pertaining to the geological environment described above,the numerical calculation model was oversimplifed and support theory underdeveloped;therefore,it is imperative to develop an efective support system for the stability and sustenance of deep roadways.In this study,a quantitative analysis of the geological environment of the roadway through feld observations,borehole-scoping,and ground stress testing is carried out to establish the FLAC 3D variable cross-section crossing roadway model.This model is combined with the strain softening constitutive(surrounding rock)and Mohr–Coulomb constitutive(other deep rock formations)models to construct a compression arch mechanical model for deep soft rock,based on the quadratic parabolic Mohr criterion.An integrated control technology of bolting and grouting that is mainly composed of a high-strength hollow grouting cable bolt equipped with modifed cement grouting materials and a high-elongation cable bolt is developed by analyzing the strengthening properties of the surrounding rock before and after bolting,based on the Heok-Brown criterion.As a result of on-site practice,the following conclusions are drawn:(1)The plastic zone of the roof of the cross roadway is approximately 6 m deep in this environment,the tectonic stress is nearly 30 MPa,and the surrounding rock is severely fractured.(2)The deformation of the roadway progressively increases from small to large cross-sections,almost doubling at the largest cross-section.The plastic zone is concentrated at the top plate and shoulder and decreases progressively from the two sides to the bottom corner.The range of stress concentration at the sides of the intersection roadway close to the passageway is wider and higher.(3)The 7 m-thick reinforced compression arch constructed under the strengthening support scheme has a bearing capacity enhanced by 1.8 to 2.3 times and increase in thickness of the bearing structure by 1.76 times as compared to the original scheme.(4)The increase in the mechanical parameters c andφof the surrounding rock after anchoring causes a signifcant increase inσt;the pulling force of the cable bolt beneath the new grouting material is more than twice that of ordinary cement grout,and according to the test,the supporting stress feld shows that the 7.24 m surrounding rock is compacted and strengthened in addition to providing a strong foundation for the bolt(cable).On-site monitoring shows that the 60-days convergence is less than 30 mm,indicating that the stability control of the roadway is successful.
文摘In this paper two-dimensional differential quadrature method has been used to analyze thick Functionally Graded(FG)rotating disks with non-uniform boundary conditions and variable thickness.Material properties vary continuously along both radial and axial directions by a power law pattern.Three-dimensional solid mechanics theory is employed to formulate the axisymmetric problem as a second order system of partial differential equations.The non-uniform boundary conditions are exerted directly into the governing equations to reach the eigenvalue system of equations.Four different disk profile shapes are considered and discussed.The effect of the power law exponent is also investigated and results show that by the use of material which functionally varied along the radial and especially axial directions the stresses and strains can be controlled so the capability of the disk is increased.Comparison with other available approaches in the literature shows a good agreement here in terms of computational time,robustness and accuracy of the present method.Moreover,novel applications are shown to provide results for further studies on the same topics.
文摘This article presents the investigation of nonlinear vibration analysis of tapered porous functionally graded skew(TPFGS)plate considering the effects of geometrical non-uniformities to optimize the thickness in the structural design.The TPFGS plate is analyzed considering linearly,bi-linearly,and exponentially varying thicknesses.The plate’s effective material properties are tailor-made using a modified power-law distribution in which gradation varies along the thickness direction of the TPFGS plate.Incorporating the non-linear finite element formulation to develop the kinematic equation’s displacement model for the TPFGS plate is based on the first-order shear deformation theory(FSDT)in conjunction with von Karman’s nonlinearity.The nonlinear governing equations are established by Hamilton’s principle.The direct iterative method is adopted to solve the nonlinear mathematical relations to obtain the nonlinear frequencies.The influence of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the TPFGS plate for different skew angles and variable thicknesses are studied for various geometrical parameters.The influence of taper ratio,variable thickness,skewness,porosity distributions,gradation,and boundary conditions on the plate’s nonlinear vibration is demonstrated.The nonlinear frequency analysis reveals that the geometrical nonuniformities and porosities significantly influence the porous functionally graded plates with varying thickness than the uniform thickness.Besides,exponentially and linearly variable thicknesses can be considered for the thickness optimizations of TPFGS plates in the structural design.
