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.

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.

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.

Suppression of multiple modal resonances of a cantilever beam by an impact damper

Impact dampers are usually used to suppress single mode resonance. The goal of this paper is to clarify the difference when the impact damper suppresses the resonances of different modes. A cantilever beam equipped with the impact damper is modeled. The elastic contact of the ball and the cantilever beam is described by using the Hertz contact model. The viscous damper between the ball and the cantilever beam is modeled to consume the vibrational energy of the cantilever beam. A piecewise ordinary differential-partial differential equation of the cantilever beam is established, including equations with and without the impact damper. The vibration responses of the cantilever beam with and without the impact damper are numerically calculated. The effects of the impact absorber parameters on the vibration reduction are examined. The results show that multiple resonance peaks of the cantilever beam can be effectively suppressed by the impact damper. Specifically, all resonance amplitudes can be reduced by a larger weight ball. Moreover, the impacting gap is very effective in suppressing the vibration of the cantilever beam. More importantly, there is an optimal impacting gap for each resonance mode of the cantilever beam, but the optimal gap for each mode is different.

BIPARAMETRIC PERTURBATION SOLUTIONS OF LARGE DEFLECTION PROBLEM OF CANTILEVER BEAMS

The large deflection problem of cantilever beams was studied by means of the biparametric perturbation method and the first order derivative substitution from pseudolinear analysis approach. This kind of substitution can transform the basic equation, an integral differential equation into nonlinear algebraic ones, thus simplify computational process. Compared with present results, it indicates that the large deflection problem solved by using pseudolinear analysis can lead to simple and precise results.

Four-Beam Model for Vibration Analysis of a Cantilever Beam with an Embedded Horizontal Crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton＇s principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.

Measurement of length-scale and solution of cantilever beam in couple stress elasto-plasticity

Owing to the absence of proper analytical solution of cantilever beams for couple stress/strain gradient elasto-plastic theory, experimental studies of the cantilever beam in the micro-scale are not suitable for the determination of material length-scale. Based on the couple stress elasto-plasticity, an analytical solution of thin cantilever beams is firstly presented, and the solution can be regarded as an extension of the elastic and rigid-plastic solutions of pure bending beam. A comparison with numerical results shows that the current analytical solution is reliable for the case of σ0 〈〈 H 〈〈 E, where σ0 is the initial yield strength, H is the hardening modulus and E is the elastic modulus. Fortunately, the above mentioned condition can be satisfied for many metal materials, and thus the solution can be used to determine the material length-scale of micro-structures in conjunction with the experiment of cantilever beams in the micro-scale.

CRACK IDENTIFICATION IN STEPPED CANTILEVER BEAM COMBINING WAVELET ANALYSIS WITH TRANSFORM MATRIX

This paper illustrates the crack identification method combining wavelet analysis with transform matrix. Firstly, the fundamental vibration mode was applied to wavelet analysis. The crack location was found by the peaks of the wavelet coefficients. Secondly, based on the identified crack locations, a simple transform matrix method requiring only the first two tested natural frequencies was used to further identify the crack depth. The present method can be used for crack identification in a complex structure. Numerical results of crack identification of a stepped cantilever beam show that the suggested method is feasible.

Approximate analytical solutions and experimental analysis for transient response of constrained damping cantilever beam

Vibration mode of the constrained damping cantilever is built up according to the mode superposition of the elastic cantilever beam. The control equation of the constrained damping cantilever beam is then derived using Lagrange＇s equation. Dynamic response of the constrained damping cantilever beam is obtained according to the principle of virtual work, when the concentrated force is suddenly unloaded. Frequencies and transient response of a series of constrained damping cantilever beams are calculated and tested. Influence of parameters of the damping layer on the response time is analyzed. Analyitcal and experimental approaches are used for verification. The results show that the method is reliable.

Dynamic analysis of a deployable/retractable damped cantilever beam

Deployable/retractable damped cantilever beams are a class of time-varying parametric structures which have attracted considerable research interest due to their many potential applications in the intelligent robot field and aerospace.In the present work,the dynamic characteristics of a deployable/retractable damped cantilever beam are investigated experimentally and theoretically.The time-varying damping,as a function of the beam length,is obtained by both the enveloped fitting method and the period decrement method.Furthermore,the governing equation of the deployable/retractable damped cantilever beam is derived by introducing the time-varying damping parameter,and the corresponding closed-form solution and vibration principles are investigated based on the averaged method.The theoretical predictions for transient dynamic responses are in good agreement with the experimental results.The dynamic mechanism analysis on time-varying damping offers flexible technology in mechanical and aerospace fields.

Analytical solution for functionally graded anisotropic cantilever beam under thermal and uniformly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to thermal and uniformly dis-tributed load is investigated,with material parameters being arbitrary functions of the thickness coordinate. The heat conduction problem is treated as a 1D problem through the thickness. Based on the elementary formulations for plane stress problem,the stress function is assumed to be in the form of polynomial of the longitudinal coordinate variable,from which the stresses can be derived. The stress function is then determined completely with the compatibility equation and boundary conditions. A practical example is presented to show the application of the method.

STATIC STUDY OF CANTILEVER BEAM STICTION UNDER ELECTROSTATIC FORCE INFLUENCE

The model and analysis of the cantilever beam adhesion problem under the action of electrostatic force are given. Owing to the nonlinearity of electrostatic force, the analytical solution for this kind of problem is not available. In this paper, a systematic method of generating polynomials which are the exact beam solutions of the loads with di?erent distributions is provided. The polynomials are used to approximate the beam displacement due to electrostatic force. The equilibrium equation o?ers an answer to how the beam deforms but no information about the unstuck length. The derivative of the functional with respect to the unstuck length o?ers such information. But to compute the functional it is necessary to know the beam deformation. So the problem is iteratively solved until the results are converged. Galerkin and Newton-Raphson methods are used to solve this nonlinear problem. The e?ects of dielectric layer thickness and electrostatic voltage on the cantilever beam stiction are studied. The method provided in this paper exhibits good convergence. For the adhesion problem of cantilever beam without electrostatic voltage, the analytical solution is available and is also exactly matched by the computational results given by the method presented in this paper.

The analytical solutions for orthotropic cantilever beams (Ⅰ):Subjected to surface forces

This paper first gives the general solution of two-dimensional orthotropic media expressed with two harmonic displacement functions by using the governing equations. Then, based on the general solution in the case of distinct eigenvalues, a series of beam problems, including the problem of cantilever beam under uniform loads, cantilever beam with axial load and bending moment at the free end, cantilever beam under the first, second, third and fourth power ofx tangential loads, is solved by the superposition principle and the trial-and-error methods.

The analytical solutions for orthotropic cantilever beams (II):Solutions for density functionally graded beams

In this paper, the specific solutions of orthotropic plane problems with body forces are derived. Then, based on the general solution in the case of distinct eigenvalues and the specific solution for density functionally graded orthotropic media, a series of beam problem, including the problems of cantilever beam with body forces depending only on z or on x coordinate and expressed by z or x polynomial is solved by the principle of superposition and the trial-and-error method.

ACTIVE VIBRATION CONTROL OF A CANTILEVER BEAM USING PIEZOELECTRIC SENSORS AND ACTUATORS

讨论了利用压电片作为传感器和驱动器对悬臂梁进行主动振动控制问题，针对模态频率差距较大的系统，提出了分组建模和控制的方法。使用滤波器消除观测溢出。文中还讨论了传感器／驱动器配置问题。实验结果验证了所述方法的有效性。

The theoretical analysis of dynamic response on cantilever beam of variable stiffness

The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable coefficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients becomes that with constant coefficients by variable substitution. At last,the study obtains the solution of dynamic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant and that at clamped end is varied,the dynamic characteristics are analyzed under several cases. The results demonstrate that the natural angular frequency reduces as the flexural rigidity reduces. When the rigidity of clamped end is higher than that of free end,low-level mode contributes the larger displacement response to the total response. On the contrary,the contribution of low-level mode is lesser than that of high-level mode.

NONLINEAR DYNAMIC MODELING AND PERIODIC VIBRATION OF A CANTILEVER BEAM SUBJECTED TO AXIAL MOVEMENT OF BASEMENT

Dynamic modeling of a cantilever beam under an axial movement ofits basement is present- ed. The dynamic equation of motion for thecantilever beam is established by using Kane's equation first andthen simplified through the Rayleigh-Ritz method. Compared with oldermodeling method, which lineari- zes the generalized inertia forcesand the generalized active forces, the present modeling takes thecoupled cubic nonlinearities of geometrical and inertial types intoconsideration.

ELASTIC EFFECT IN DYNAMIC RESPONSE OF PLASTIC CANTILEVER BEAMS TO IMPACT

A lumped masses-springs model is proposed to analyze the dynamic response of an elastic-plastic cantilever beam resulting from impact. The numerical results are in good agreement with those by finite-element approaches. The simplified model can catch the most essential features of elastic-plastic response of beams; in particular, it demonstrates the effect of elastic deformation on the distribution of bending moment and energy dissipation, and provides valuable quatitative results.

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magnetomelectro-elastic （FGMEE） cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.