Theory of Flexural Shear, Bending and Torsion for a Thin-Walled Beam of Open Section

Aspects of the general Vlasov theory are examined separately as applied to a thin-walled channel section cantilever beam under free-end end loading. In particular, the flexural bending and shear that arise under transverse shear and axial torsional loading are each considered theoretically. These analyses involve the location of the shear centre at which transverse shear forces when applied do not produce torsion. This centre, when taken to be coincident with the centre of twist implies an equivalent reciprocal behaviour. That is, an axial torsion applied concentric with the shear centre will twist but not bend the beam. The respective bending and shear stress conversions are derived for each action applied to three aluminium alloy extruded channel sections mounted as cantilevers with a horizontal principal axis of symmetry. Bending and shear are considered more generally for other thin-walled sections when the transverse loading axes at the shear centre are not parallel to the section = s centroidal axes of principal second moments of area. The fixing at one end of the cantilever modifies the St Venant free angular twist and the free warping displacement. It is shown from the Wagner-Kappus torsion theory how the end constrained warping generates an axial stress distribution that varies with the length and across the cross-section for an axial torsion applied to the shear centre. It should be mentioned here for wider applications and validation of the Vlasov theory that attendant papers are to consider in detail bending and torsional loadings applied to other axes through each of the centroid and the web centre. Therein, both bending and twisting arise from transverse shear and axial torsion applied to each position being displaced from the shear centre. Here, the influence of the axis position upon the net axial and shear stress distributions is to be established. That is, the net axial stress from axial torsional loading is identified with the sum of axial stress due to bending and axial stress arising from constrained warping displacements at the fixing. The net shear stress distribution overlays the distributions from axial torsion and that from flexural shear under transverse loading. Both arise when transverse forces are displaced from the shear centre.

GEOMETRICALLY NONLINEAR FINITE ELEMENT MODEL OF SPATIAL THIN-WALLED BEAMS WITH GENERAL OPEN CROSS SECTION

Based on the theory of Timoshenko and thin-walled beams, a new finite element model of spatial thin-walled beams with general open cross sections is presented in the paper, in which several factors are included such as lateral shear deformation, warp generated by nonuni- form torsion and second-order shear stress, coupling of flexure and torsion, and large displacement with small strain. With an additional internal node in the element, the element stiffness matrix is deduced by incremental virtual work in updated Lagrangian （UL） formulation. Numerical examples demonstrate that the presented model well describes the geometrically nonlinear property of spatial thin-walled beams.

Modeling and Free Vibration Behavior of Rotating Composite Thin-walled Closed-section Beams with SMA Fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy（SMA） fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton＇s principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin＇s method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.

Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

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.

Chebyshev polynomial-based Ritz method for thermal buckling and free vibration behaviors of metal foam beams

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.

Deformation Analysis and Fixture Design of Thin-walled Cylinder in Drilling Process Based on TRIZ Theory

Thin-walled cylindrical workpiece is easy to deform during machining and clamping processes due to the insufficient rigidi.Moreover,it’s also difficult to ensure the perpendicularity of flange holes during drilling process.In this paper,the element birth and death technique is used to obtain the axial deformation of the hole through finite element simulation.The measured value of the perpendicularity of the hole was compared with the simulated value to verify then the rationality of the simulation model.To reduce the perpendicularity error of the hole in the drilling process,the theory of inventive principle solution(TRIZ)was used to analyze the drilling process of thin-walled cylinder,and the corresponding fixture was developed to adjust the supporting surface height adaptively.Three different fixture supporting layout schemes were used for numerical simulation of drilling process,and the maximum,average and standard deviation of the axial deformation of the flange holes and their maximum hole perpendicularity errors were comparatively analyzed,and the optimal arrangement was optimized.The results show that the proposed deformation control strategy can effectively improve the drilling deformation of thin-walled cylindrical workpiece,thereby significantly improving the machining quality of the parts.

Intensity correlation properties of x-ray beams split with Laue diffraction

Beam splitting is one of the main approaches to achieving x-ray ghost imaging, and the intensity correlation between diffraction beam and transmission beam will directly affect the imaging quality. In this paper, we investigate the intensity correlation between the split x-ray beams by Laue diffraction of stress-free crystal. The analysis based on the dynamical theory of x-ray diffraction indicates that the spatial resolution of diffraction image and transmission image are reduced due to the position shift of the exit beam. In the experimental setup, a stress-free crystal with a thickness of hundredmicrometers-level is used for beam splitting. The crystal is in a non-dispersive configuration equipped with a double-crystal monochromator to ensure that the dimension of the diffraction beam and transmission beam are consistent. A correlation coefficient of 0.92 is achieved experimentally and the high signal-to-noise ratio of the x-ray ghost imaging is anticipated.Results of this paper demonstrate that the developed beam splitter of Laue crystal has the potential in the efficient data acquisition of x-ray ghost imaging.

Dynamic and electrical responses of a curved sandwich beam with glass reinforced laminate layers and a pliable core in the presence of a piezoelectric layer under low-velocity impact

The dynamic responses and generated voltage in a curved sandwich beam with glass reinforced laminate(GRL)layers and a pliable core in the presence of a piezoelectric layer under low-velocity impact(LVI)are investigated.The current study aims to carry out a dynamic analysis on the sandwich beam when the impactor hits the top face sheet with an initial velocity.For the layer analysis,the high-order shear deformation theory(HSDT)and Frostig's second model for the displacement fields of the core layer are used.The classical non-adhesive elastic contact theory and Hunter's principle are used to calculate the dynamic responses in terms of time.In order to validate the analytical method,the outcomes of the current investigation are compared with those gained by the experimental tests carried out by other researchers for a rectangular composite plate subject to the LVI.Finite element(FE)simulations are conducted by means of the ABAQUS software.The effects of the parameters such as foam modulus,layer material,fiber angle,impactor mass,and its velocity on the generated voltage are reviewed.

Nonlinear Free Vibration Analysis of Thin-walled Curved Beam with Non-symmetric Open Cross Section

A finite element formulation was presented for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies were derived by the virtual principle. The energy function includes the effect of fiexural-torsional coupling, the torsion warping and the shear centre location. For finite element analysis, cubic polynomials were utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem was solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.

A new finite element of spatial thin-walled beams

Based on the theories of Timoshenko＇s beams and Vlasov＇s thin-walled members, a new spatial thin-walled beam element with an interior node is developed. By independently interpolating bending angles and warp, factors such as transverse shear deformation, torsional shear deformation and their Coupling, coupling of flexure and torsion, and second shear stress are considered. According to the generalized variational theory of Hellinger-Reissner, the element stiffness matrix is derived. Examples show that the developed model is accurate and can be applied in the finite element analysis of thinwalled structures.

Elastic buckling analysis and optimization of thin-walled bamboo-like pipe beam subjected to pure bending

In this paper,a stiffening design imitating the bamboo node is proposed for weight reduction of the long composite pipe beam subjected to bending load.The distribution of bamboo nodes can efficiently suppress the ovalization of the section,thus significantly improving the bending resistance of the bamboo.Based on this principle,ring stiffeners are proposed to be fixed to the pipe beam,making the long beam equivalent to the combination of a series of short pipes that suffered less section ovalization.A database of the optimal laminate orientations for different normalized lengths is obtained through optimizations,where the discreteness of the ply count is considered.Based on this database,weight optimizations are conducted,and the optimal designs of beams with and without stiffeners are obtained and compared.The comparison results show that the proposed bamboo-like stiffened beam not only regains a near-linear load–displacement relationship,but also reduces the weight by up to 16%under the same buckling load.In addition,it is found that for the pipe beams with radius-to-thickness ratios of more than 18,increasing the radius leads to a decrease in elastic buckling resistance when the weight remains a constant,which is opposite to the design for strength and stiffness.The model and database developed in this paper can provide a reference for weight reduction design and weight estimation for composite pipe beams.

Longwall mining “cutting cantilever beam theory” and 110 mining method in China——The third mining science innovation

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.

Free vibration analysis of functionally graded material beams based on Levinson beam theory

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.

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.

Optimal design of sub-wavelength metal rectangular gratings for polarizing beam splitter based on effective medium theory

A novel optimal design of sub-wavelength metal rectangular gratings for the polarizing beam splitter （PBS） is proposed. The method is based on effective medium theory and the method of designing single layer antireflection coating. The polarization performance of PBS is discussed by rigorous couple-wave analysis （RCWA） method at a wavelength of 1550 nm. The result shows that sub-wavelength metal rectangular grating is characterized by a high reflectivity, like metal films for TE polarization, and high transmissivity, like dielectric films for TM polarization. The optimal design accords well with the results simulated by RCWA method.

Optimization approach hydroforming car beam billets based grey system theory

Perfect combination of structural size parameters of the hydroforming billets is essential to obtain even wall thicknesses of the car beam. Finite element （ FE ） analysis on hydroforming car beam was carried out, and the results were optimized according to multiple quality objectives by the grey system theory. With bending angle, bending radius and hight difference along the axis direction as variables, orthogonal FE analyses were conducted and the minimum and maximum wall thicknes ses of the billets with different sizes were obtained. Taking the minimum and maximum wall thick nesses as two references, the correlation coefficient between the data for reference and those for comparison by the grey system theory reduced multi objectives to a single quality objective, and the average correlation level of every billet facilitated the optimization of size parameters for hydroform ing car beam. The trial production showed that the optimization approach satisfied the need of hy droforming car beams.

Conversion between solid and beam element solutions of finite element method based on meta-modeling theory:development and application to a ramp tunnel structure

In this study, a new method for conversion of solid finite element solution to beam finite element solution is developed based on the meta-modeling theory which constructs a model consistent with continuum mechanics. The proposed method is rigorous and efficient compared to a typical conversion method which merely computes surface integration of solid element nodal stresses to obtain cross-sectional forces. The meta-modeling theory ensures the rigorousness of proposed method by defining a proper distance between beam element and solid element solutions in a function space of continuum mechanics. Results of numerical verification test that is conducted with a simple cantilever beam are used to find the proper distance function for this conversion. Time history analysis of the main tunnel structure of a real ramp tunnel is considered as a numerical example for the proposed conversion method. It is shown that cross-sectional forces are readily computed for solid element solution of the main tunnel structure when it is converted to a beam element solution using the proposed method. Further, envelopes of resultant forces which are of primary importance for the purpose of design, are developed for a given ground motion at the end.

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

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.

An R(x)-orthonormal theory for the vibration performance of a non-smooth symmetric composite beam with complex interface

A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex nonsmooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function. The R(x)-orthonormal bases in the linear space of boundary functions are constructed, of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x). When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix A :=[aij ],aij= 0 if i + j is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of A can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions half lower. Hence, we can sequentially solve the natural frequencies in closed-form owing to the specialty of A. We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.

BENDING THEORIES FOR BEAMS AND PLATES WITH SINGLE GENERALIZED DISPLACEMENT

Classical bending theories for beams and plates can not be used for short, stubby beams and thick plates since transverse shearing effect is excluded, and ordinary theories with multiple generalized displacements can not be used for long, slender beams and thin plates since the innate relation between rotation angle and deflection is ignored. These two types of theories are not consistent due to the contradiction of dependence and independence of the rotation angle. Based on several basic assumptions, a new type of theories which not only include the transverse shearing effect is presented, but also the relation between potation angle and deflection is obtained. Analytical solutions of several simple beams are given. It has been testified by numerical examples that the new theories can be used for either long, slender beams and thin plates or short, stubby beams and thick plates.