Vibration and buckling of a double-beam system under compressive axial loading

On the basis of the Bernoulli–Euler beam theory, the properties of free transverse vibration and buckling of a double-beam system under compressive axial loading are investigated in this paper. It is assumed that the two beams of the system are simply supported and continuously joined by a Winkler elastic layer. Explicit expressions are derived for the natural frequencies and the associated amplitude ratios of the two beams, and the analytical solution of the critical buckling load is obtained. The influences of the compressive axial loading on the responses of the double-beam system are discussed. It is shown that the critical buckling load of the system is related to the axial compression ratio of the two beams and the Winkler elastic layer, and the properties of free transverse vibration of the system greatly depend on the axial compressions.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads

In this paper, a boundary element method is developed for the geometrically nonlinear response of shear deformable beams of simply or multiply connected constant cross-section, traversed by moving loads, resting on tensionless nonlinear three-parameter viscoelastic foundation, undergoing moderate large deflections under general boundary conditions. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse moving loading as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the transverse displacement, to the axial displacement and to a stress functions and solved using the Analog Equation Method, a Boundary Element based method. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which is solved using an efficient time discretization scheme, from which the transverse and axial displacements are computed. The evaluation of the shear deformation coefficient is accomplished from the aforementioned stress function using only boundary integration. Analyses are performed to illustrate, wherever possible, the accuracy of the developed method, to investigate the effects of various parameters, such as the load velocity, load frequency, shear deformation, foundation nonlinearity, damping, on the beam displacements and stress resultants and to examine how the consideration of shear and axial compression affects the response of the system.     

轴压和扭转复合载荷作用下氮化硼纳米管屈曲行为的分子动力学模拟

采用分子动力学方法模拟了氮化硼纳米管在轴压和扭转复合荷载作用下的屈曲和后屈曲行为.在各加载比例下,给出了初始线性变形阶段和后屈曲阶段原子间相互作用力的变化,确定了屈曲临界荷载关系.通过对屈曲模态的细致研究,从微观变形机理上分析了纳米管对不同外荷载力学响应的差异.研究结果表明,扶手型和锯齿型纳米管均呈现出非线性的屈曲临界荷载关系,复合加载下的屈曲行为具有强烈的尺寸依赖性.温度升高将导致屈曲临界荷载的下降,且温度的影响随加载比例的变化而变化.无论在简单加载或复合加载中,同尺寸的碳纳米管均比氮化硼纳米管具有更强地抵抗屈曲荷载的能力.     

VIBRATION AND BUCKLING OF INITIALLY STRESSED CURVED BEAMS

Equations of motion for curved beams in a general state of non-uniform initial stresses are derived using the principle of virtual work. The equations are adjusted to a generic expression by using appropriate transformations. The free vibration behaviours of the curved beams subjected to a combination of uniform initial tensile of compressive stresses and uniform initial bending stress are examined. The Galerkin method is employed in obtaining accurate values of free frequencies and initial buckling stresses. The curved beam is assumed to be vibrating in its plane. Natural frequencies and initial buckling stresses for hinged supported curved beams are presented for validation. Effects of arc segment angles, elastic foundation, and initial stresses on the natural frequencies are investigated. Effects of arc segment angles, elastic foundation, and initial bending stresses on the initial buckling stresses are explored in this paper.     

Atomistic finite element model for axial buckling and vibration analysis of single-layered graphene sheets

In this article, an atomistic model is developed to study the buckling and vibration characteristics of single-layered graphene sheets (SLGSs). By treating SLGSs as space-frame structures, in which the discrete nature of graphene sheets is preserved, they are modeled using three-dimensional elastic beam elements for the bonds. The elastic moduli of the beam elements are determined via a linkage between molecular mechanics and structural mechanics. Based on this model, the critical compressive forces and fundamental natural frequencies of single-layered graphene sheets with different boundary conditions and geometries are obtained and then compared. It is indicated that the compressive buckling force decreases when the graphene sheet aspect ratio increases. At low aspect ratios, the increase of aspect ratios will result in a significant decrease in the critical buckling load. It is also indicated that increasing aspect ratio at a given side length results in the convergence of buckling envelops associated with armchair and zigzag graphene sheets. The influence of boundary conditions will be studied for different geometries. It will be shown that the influence of boundary conditions is not significant for sufficiently large SLGSs.     

An analytical method for free vibration analysis of functionally graded beams with edge cracks

In this paper, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation. The governing differential equations of motion for an FGM beam are established and the corresponding solutions are found first. The discontinuity of rotation caused by the cracks is simulated by means of the rotational spring model. Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams. The main advantage of the proposed method is that the eigenvalue equation for vibrating beams with an arbitrary number of cracks can be conveniently determined from a third-order determinant. Due to the decrease in the determinant order as compared with previous methods, the developed method is simpler and more convenient to analytically solve the free vibration problem of cracked FGM beams. Moreover, free vibration analyses of the Euler–Bernoulli and Timoshenko beams with any number of cracks can be conducted using the unified procedure based on the developed method. These advantages of the proposed procedure would be more remarkable as the increase of the number of cracks. A comprehensive analysis is conducted to investigate the influences of the location and total number of cracks, material properties, axial load, inertia and end supports on the natural frequencies and vibration mode shapes of FGM beams. The present work may be useful for the design and control of damaged structures.     

Vibration and stability of elastically supported beams carrying an attached mass under axial and tangential loads

The vibration and stability of an elastically supported beam carrying an attached mass and subjected to axial and tangential compressive loads are investigated. The analysis is based on the Timoshenko beam theory and the effects of the attached mass are expressed with Dirac delta functions. The influences of the support stiffness, the direction of loading, and the slenderness ratio on the natural frequency and critical load of a beam are discussed.     

Effects of structural defects on the compressive buckling of boron nitride nanotubes

In this paper, we examined the buckling of perfect and defective armchair boron nitride nanotubes with three types of vacancy defects, i.e. B- and N- single vacancy defects and B–N- double vacancy defect, using molecular dynamics simulations. To this end, all systems were modeled with a Tersoff-type potential, which is able to accurately describe covalent bonding of BN systems. We applied external uniaxial compressive forces to the nanotubes in vacuum and derived the critical buckling loads and strains, at room temperature in an NVT-ensemble. Our results showed significant differences between the critical buckling strengths of pristine and defective nanotubes. The resistance to axial buckling decreased with the introduction of one vacancy defect, and the B–N- double vacancy was the most seriously damaged structure, followed by B-vacancy and N-vacancy defects. Furthermore, the B-vacancy was shown to have the most significant effect on the decrease of the critical buckling strain. This can be attributed to the excessive asymmetries and perturbations induced in the structure of the nanotube and the local deformations around the defective site around the B-vacancy, even before loading. Moreover, results show that reduction in the buckling strength of the nanotube due to the presence of more than one B-vacancy defect depends on their distribution. If the two or three defects are close to each other, they act as a single point of weakness and the critical buckling load is only slightly reduced (similar to the existence of only one vacancy defect). However, if the defects are at more distant points, the critical buckling load may experience a higher decrease. Results show that vacancy defects play a critical role in the compressive buckling performance of boron nitride nanotubes and special attention must be paid to the presence of structural defects when designing members against buckling, especially for micro- and nano-electro-mechanical systems. On the other hand, defect engineering is a great means for tailoring the buckling strength of boron nitride nanotubes, in cases where the nanotube is expected to absorb energy through compressive buckling deformation and is not designed against, but for buckling.     

Vibration of a two-member open frame

The dynamics of a two-member open frame structure undergoing both in- and out-of-plane motion is examined. The frames are modelled using the Euler-Bernoulli beam theory and are further generalized by permitting an arbitrary angle between the beams and the attachment of a payload at the end of the second beam. The equations of motion are derived using Hamilton's principle and the orthogonality conditions are presented. It is shown that the in- and out-of-plane motions can be decoupled by including the axial deformation components in the assumed displacement fields. The natural frequencies of the system and the contribution of each member into the system potential energy are examined via numerical examples.     

Buckling analysis of carbon nanotube bundles under axial compressive,bending and torsional loadings via a structural mechanics model

A structural mechanics model is employed for the investigation of the buckling behavior of carbon nanotube bundles of three single-walled carbon nanotubes (SWCNTs) under axial compressive, bending and torsional loadings. The effects of van der Waals (vdW) forces are further modeled using a nonlinear spring element.The effects of different types of boundary conditions are studied for nanotubes with various aspect ratios. The results reveal that bundles comprising longer SWCNTs exhibit lower critical buckling load. Moreover, for the fixed-free boundary condition the rate of critical buckling load reduction is highest, while the lowest critical buckling load occurs. Simulations show good agreement between our model and molecular dynamics results.     

Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects

The dynamic transfer matrix method for determining natural frequencies and mode shapes of the bending-torsion coupled vibration of axially loaded thin-walled beams with monosymmetrical cross sections is developed by using a general solution of the governing differential equations of motion based on Bernoulli-Euler beam theory. This method takes into account the effect of warping stiffness and gives allowance to the presence of axial force. The dynamic transfer matrix is derived in detail. Two illustrative examples on the application of the present theory are given for bending-torsion coupled beams with thin-walled open cross sections. The effects of axial load and warping stiffness on coupled bending-torsional frequencies are discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the proposed method.     

Coupled flexural–longitudinal vibration of delaminated composite beams with local stability analysis

A novel analytical model is developed to solve the problem of free vibration of delaminated composite beams. The beam with a single delamination was modelled by six equivalent single layers by establishing the kinematic continuity in the undelaminated portion of the system. In the delaminated region the layers were captured by the traditional theories. First, Timoshenko beam theory is applied to solve the problem, then by reducing the model, the corresponding Euler–Bernoulli solution is presented. Both the free and constrained models were considered. The most important aspect of the present analysis is that the beams of the delaminated region are subjected to normal forces, as well. That is the essential reason for leading to a coupled flexural–longitudinal vibration problem. It is also concluded that delamination buckling can take place if the normal force is compressive in one of the half-periods of the vibration and reaches a critical value. The problem was also investigated experimentally by modal hammer and sweep excitation tests on beams made of E-glass/polyester in order to measure the natural frequencies and mode shapes. The comparison of the analytical and experimental results indicates the importance of the independent rotations provided by Timoshenko beams over the simple beam theory. The delamination buckling of the beams was captured based on the static stability analysis in the first step. Further results show that the problem is more complex than it was thought before, e.g., some nonlinearity, time-dependent stiffness as well as parametric excitation aspects were discovered during the present analysis.     

定跨度变截面梁的弯曲问题

一.引言 在处理梁的弯曲问题时,人们经常利用函数级数来表示有关的各量,并后而得到各该量的近似值。胡海昌曾经指出:在横向载荷和轴向力同时作用下,适宜于用梁的屈曲的本徵函数展开式来表示梁的挠度;其中φ_n是满足所给的梁的支座情况的屈曲本徵函数,a_n是常数系数。他求得一个相当简单的公式以已知的本徵函数和本徵值表示诸系数     

Crack effects on the in-plane static and dynamic stabilities of a curved beam with an edge crack

The effects of a single-edge crack and its locations on the buckling loads, natural frequencies and dynamic stability of circular curved beams are investigated numerically using the finite element method, based on energy approach. This study consists of three stages, namely static stability (buckling) analysis, vibration analysis and dynamic stability analysis. The governing matrix equations are derived from the standard and cracked curved beam elements combined with the local flexibility concept. Approximation for the displacements using coupled interpolations based on the constant-strain, linear-curvature element (SC) has yielded results with reasonable accuracy. The numerical results obtained from the present finite element model are found to be in good agreement with those, both experimental and analytic, of other researchers in the existing literature. Results show that the reductions in buckling load and natural frequency depend not only on the crack depth and crack position, but also on the related mode shape. Analyses also show that the crack effect on the dynamic stability of the considered curved beam is quite limited.     

Size effect in transverse mechanical behavior of one-dimensional nanostructures

This paper develops the classical strain gradient elasticity theory to investigate the scale-dependent mechanical behavior of one-dimensional (1D) nanostructures. A governing differential equation with two scale parameters is derived, where the curvature of the deflection and the higher-order bending moment are introduced as a pair of additional geometrical constraint and natural loading. Emphasis is placed on the analysis of bending deformation, free vibration and buckling of cantilever nanowires or free-standing nanocolumns. Obtained results are compared with experimental data of carbon nanotube ropes and nanowires available in the literature and they agree well, showing that transverse mechanical properties of nanowires such as bending stiffness are scale-dependent. The model proposed also indicates that the evaluated natural frequencies and critical buckling strains exhibit noticeable size effects. Bending stiffness, natural frequency and buckling load increase as the nanowire diameter drops down. The influence of rotary inertia of cross-section is also analyzed.     

Column buckling of doubly parallel slender nanowires carrying electric current acted upon by a magnetic field

Axial buckling of current-carrying double-nanowire-systems immersed in a longitudinal magnetic field is aimed to be explored. Each nanowire is affected by the magnetic forces resulted from the externally exerted magnetic field plus the magnetic field resulted from the passage of electric current through the adjacent nanowire. To study the problem, these forces are appropriately evaluated in terms of transverse displacements. Subsequently, the governing equations of the nanosystem are constructed using Euler–Bernoulli beam theory in conjunction with the surface elasticity theory of Gurtin and Murdoch. Using a meshless technique and assumed mode method, the critical compressive buckling load of the nanosystem is determined. In a special case, the obtained results by these two numerical methods are successfully checked. The roles of the slenderness ratio, electric current, magnetic field strength, and interwire distance on the axial buckling load and stability behavior of the nanosystem are displayed and discussed in some detail.     

Studying the buckling and vibration characteristics of single-walled zinc oxide nanotubes using a nanoscale finite element model

The free vibration and axial buckling of achiral zinc oxide nanotubes (ZnONTs) are studied in this paper based on a three-dimensional finite-element model in which bonds are modeled using beam elements and mass elements are placed at the joints of beams instead of atoms. To determine the mechanical properties of the nanotubes, a linkage is established between molecular mechanics and density functional theory. The fundamental frequency and critical buckling load of ZnONTs with different geometries, chiralities and boundary conditions are calculated. It is shown that zigzag nanotubes are more stable than armchair ones. Investigating the effect of aspect ratio on the critical force shows that longer nanotubes are less stable. Also, it is indicated that increasing the length of the nanotubes will result in decreasing the frequency. Moreover, as the aspect ratio increases, the effect of end conditions diminishes.     

Stability and vibration of isotropic,orthotropic and laminated plates according to a higher-order shear deformation theory

A higher-order shear deformation theory is used to determine the natural frequencies and buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia. Exact solutions of simply supported plates are obtained and the results are compared with the exact solutions of three-dimensional elasticity theory, the first-order shear deformation theory, and the classical plate theory. The present theory predicts the frequencies and buckling loads more accurately when compared to the first-order and classical plate theories.     

MODELLING OF TRANSVERSE VIBRATION OF SHORT BEAMS FOR CRACK DETECTION AND MEASUREMENT OF CRACK EXTENSION

The method of detection of location of crack in beams based on frequency measurements is extended here to short beams, taking into account the effects of shear deformation and rotational inertia through the Timoshenko beam theory and representing the crack by a rotational spring. Methods for solving both forward (determination of frequencies of beams knowing the crack parameters) and inverse (determination of crack location knowing the natural frequencies) problems are included. A method to estimate crack extension from a change in the first natural frequency is presented. Both numerical and experimental studies are given to demonstrate the accuracy of the methods. The accuracy of the results is quite encouraging.