分析弹性支承输流管道的失稳临界流速

研究了两端弹性支承输流管道静态失稳和动态失稳临界流速. 根据梁模型横向弯曲振动模态 函数，由两端弹性支承的边界条件得到了其模态函数的一般表达式. 根据特征方程具体分析 了弹性支承刚度、质量比、流体压力和管截面轴向力等主要参数对失稳临界流速的影响. 数 值计算结果表明，管道在弹性支承下的动力稳定性比较复杂，在较小的弹性支承刚度和较小 的参数范围内，管道主要表现为动态颤振失稳；在较大的弹性支承刚度和较大的参数作用下， 管道的失稳形式主要表现为静态失稳；并且失稳临界流速随流体压力和管截面轴向压力的增 加而下降，随管截面轴向拉力的增加而上升.     

Twin-characteristic-parameter solution of axisymmetric dynamic plastic buckling for cylindrical shells under axial compression waves

In the present investigation on the dynamic plastic buckling of cylindrical shells under axial compression waves, the critical axial stress and the exponential parameter of inertia terms in stability equations are treated as a couple of characteristic parameters. The criterion of transformation and conservation of energy in the process of buckling initiation is used to derive the supplementary restraint equation of buckling deformation at the fronts of axial elastic and plastic compression waves. The supplementary restraint equation, stability equations, boundary conditions and continuity conditions constitute the necessary and sufficient conditions of determining the two characteristic parameters. Two characteristic equations are derived for the two characteristic parameters. The critical axial stress or the critical buckling time, the exponential parameter of inertia terms and the initial modes of buckling deformation are calculated quantitatively from the solution of the characteristic equations.     

Non-conservative stability of intermediate spring supported columns with an elastically restrained end support

The non-conservative stability of an intermediate spring supported uniform column clastically restrained at one end and subjected to a follower force at the other unsupported end is studied. It is found that when the intermediate spring support is far from the unsupported end, the instability mechanism is flutter. As the intermediate spring support approaches the unsupported end, the instability mechanism is changed from flutter to divergence with the increase of intermediate spring stiffness. For the hinged-intermediate and guided-intermediatc spring supported columns, the critical buckling load of flutter instability will first decrease, then increase as the intermediate spring stiffness is increased. Nevertheless, when the instability mechanism is divergence, the critical buckling load depends on the location of the intermediate spring support only, whereas for the clamped-intermediate spring supported column the critical buckling load of divergence instability decreases monotonically to a fixed value as the intermediate spring stiffness is increased. Finally, the influence of elastic end restraints on the stability of the column is also investigated.     

On the influence of the gravity force on the stability of an inhomogeneous layer under biaxial extension-compression

In the present paper, in the framework of the three-dimensional nonlinear theory of elasticity, we study the stability of a heavy layer under biaxial extension-compression. The elastic properties of the layer are assumed to be inhomogeneous along thickness and are described by a semilinear material model. We study the stability by using the bifurcation approach. By solving the linearized equilibrium equations, we obtain the critical curves and the stability domain in the plane of the loading parameters, for which we take the material elongation ratios along the coordinate axes lying in the layer plane. We analyze the influence of the layer thickness, specific weight, and material parameters on buckling. In particular, we find that, when studying stability, it is expedient to take the gravity force into account only if the layer rigidity decreases with increasing depth.     

Surface layer stability under force and temperature loading

We study stability of an elastic homogeneous ormultilayer plate with a free surface under the action of force and temperature deformations. We show that possible buckling modes can be of two types. For each of these two types, we study the dependence of the critical strain on the wavelength. It was found that, for a homogeneous layer, a decrease in the wavelength leads to a decrease in the critical strain and, for a multilayer material with alternating hard and soft layers, a strain wave of finite length arises in buckling.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

DIVERGENCE INSTABILITY OF VARIABLE-ARC-LENGTH ELASTICA PIPES TRANSPORTING FLUID

A flexible elastic pipe transporting fluid is held by an elastic rotational spring at one end, while at the other end, a portion of the pipe may slide on a frictional support. Regardless of the gravity loads, when the internal flow velocity is higher than the critical velocity, large displacements of static equilibrium and divergence instability can be induced. This problem is highly nonlinear. Based on the inextensible elastica theory, it is solved herein via the use of elliptic integrals and the shooting method. Unlike buckling with stable branching of a simply supported elastica pipe with constant length, the variable arc-length elastica pipe buckles with unstable branching. The friction at the support has an influence in shifting the critical locus over the branching point. Alteration of the flow history causes jumping between equilibrium paths due to abrupt changes of direction of the support friction. The elastic rotational restraint brings about unsymmetrical bending configurations; consequently, snap-throughs and snap-backs can occur on odd and even buckling modes, respectively. From the theoretical point of view, the equilibrium configurations could be formed like soliton loops due to snapping instability.     

Volume and surface stability of uniformly compressed transversely isotropic linearly elastic materials

The stability loss of a transversely isotropic linearly elastic medium is studied. The medium is uniformly compressed in both horizontal directions, and the initial stress in the vertical direction is equal to zero. The standard analysis based on the Hadamard condition is used. The bifurcation equation divides into two parts, and therefore, two kinds of buckling modes are possible. The critical initial compression is found, but the buckling modes remain indefinite (as the wave length so the relation between the wave numbers is arbitrary). The stability loss of a compressed half-space with a free surface is studied. Only one kind of buckling mode localized near the free surface is possible, and as for an entire space, the buckling mode and the wave length are indefinite. In these problems, linear as well as non-linear approaches are used. In the linear approach, the pre-buckling deformations are ignored. It is shown that for some values of parameters, the linear approach leads to qualitatively incorrect results. The stability loss of an uniformly compressed plate lying on a soft elastic half-space is studied. By using the non-linear post-critical analysis, it is shown that the buckling mode is a chessboard-like one.     

Stability Analysis of Maxwell Viscoelastic Pipes Conveying Fluid with Both Ends Simply Supported

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ON THE ORIENTATION OF BUCKLING DIRECTION OF ANISOTROPIC ELASTIC PLATE UNDER UNIAXIAL COMPRESSION

The theory of small deformation superimposed on a large deformation of an elastic solid is used to investigate the buckling of anisotropic elastic plate under uniaxial compression. The buckling direction (the direction of buckling wave) is generally not aligned with the compression direction. The equation for determining the buckling direction is obtained. It is found that the out-of-plane buckling of anisotropic elastic plate is possible and both buckling conditions for flexural and extensional modes are presented. As a specific case of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate subjected to a compression in a direction that forms an arbitrary angle with an elastic principal axis of the materials is analyzed. It is found that the buckling direction depends on the angle between the compression direction and the principal axis of the materials, the critical compressive force and plate-thickness parameters. In the case that the compression direction is aligned with the principal axis of the materials, the buckling direction will be aligned with the compression one irrespective of critical compressive force and plate-thickness. Project supported by the National Natural Science Foundation of China (No. 19772032).     

U型波纹管及相关结构环向屈曲的有限元分析(Ⅰ)--基本方程及环板的屈曲

U型波纹管是现代管道系统中最常见的一种位移补偿器 ,它由环板和具有正、负Gauss曲率的半圆环壳组成 ,在管道所传输的介质的压力作用下会发生屈曲。其中环向屈曲最为复杂 ,精确的理论分析非常困难 ,有限元分析也不多见。作者在分析前人工作的基础上 ,以圆环壳段为单元 (特定的旋转壳段单元 ,能自动退化成环板单元 ) ,限于弹性范围和线性化特征值问题 ,对介质压力作用下U型波纹管及其相关结构 (圆环板、圆环壳、半圆环壳 )的环向屈曲问题进行了分析。考虑了结构屈曲前的弯曲 ,计及压力的二次势能 ,导出的应力刚度矩阵和载荷刚度矩阵是非对称的。全部工作分为三部分 :(Ⅰ )基本方程 ,环板的屈曲 ;(Ⅱ )圆环壳、半圆环壳的屈曲 ;(Ⅲ )波纹管平面失稳的机理。本文为第一部分 ,除推导公式外 ,对不同边界和不同内外径之比的环板在径向均匀压力作用下的环向屈曲进行了计算 (轴对称的径向屈曲作为特例得到 ) ,给出了前屈曲应力分布、临界载荷及相应的屈曲模态 ,并将临界压力的值与前人基于vonK偄ｒｍ偄ｎ大挠度板的精确解进行了比较 ,吻合良好。     

基于三阶剪切变形理论的悬臂输流管道自由振动

采用三阶剪切变形理论,结合有限元法研究了悬臂输流管道的自由振动问题．利用虚功原理建立了输流管系统的有限元方程,同时将悬臂端弹性支承以势能的形式引入到系统方程中,求解了系统前三阶的复频率．分别探讨了流体速度和弹簧刚度对系统复频率实部和虚部的影响,重点分析了弹簧刚度与前三阶固有频率间的关系．在弹性支承刚度为零的特例下,对比了本文结果与Timoshenko梁理论的结果,证明了本文方法的可靠性．研究发现系统固有频率的实部恒为负值,表明一端带有弹性支承的约束形式有利于提高悬臂输流管道自由振动的稳定性;流体的流动对管道振动起到了阻尼作用,在流动速度足够大的情况下,各阶振动固有频率均趋于零;当弹簧刚度为无穷大,且流体速度足够大时,输流管道将发生失稳．     

Nonlinear analysis and buckling of elastically supported circular shallow arches

Arches are often supported elastically by other structural members. This paper investigates the in-plane nonlinear elastic behaviour and stability of elastically supported shallow circular arches that are subjected to a radial load uniformly distributed around the arch axis. Analytical solutions for the nonlinear behaviour and for the nonlinear buckling load are obtained for shallow arches with equal or unequal elastic supports. It is found that the flexibility of the elastic supports and the shallowness of the arch play important roles in the nonlinear structural response of the arch. The limiting shallownesses that distinguish between the buckling modes are obtained and the relationship of the limiting shallowness with the flexibility of the elastic supports is established, and the critical flexibility of the elastic radial supports is derived. An arch with equal elastic radial supports whose flexibility is larger than the critical value becomes an elastically supported beam curved in elevation, while an arch with one rigid and one elastic radial support whose flexibility is larger than the critical value still behaves as an arch when its shallowness is higher than a limiting shallowness. Comparisons with finite element results demonstrate that the analytical solutions and the values of the critical flexibility of the elastic supports and the limiting shallowness of the arch are valid.     

Torsional buckling of an elastic thick-walled tube made of rubber-like material

The elastic stability of a rubber-like, thick-walled tube which is subjected to finite torsional deformation is investigated both theoretically and experimentally. The analysis is based on the theory of finite elastic deformations, in cojunction with the method of small displacements superposed on large elastic deformations. The governing field equations are solved by a numerical scheme which determines the critical buckling torque and the associated buckling mode of the tube. The predicted results compare closely with the experimental measurements of the buckling of thick-walled silicone rubber tubes tested under finite twist.     

The thermal effect on axially compressed buckling of a double-walled carbon nanotube

The thermal effect on axially compressed buckling of a double-walled carbon nanotube is studied in this paper. The effects of temperature change, surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model with thermal effect is presented for axially compressed buckling of a double-walled carbon nanotube embedded in an elastic matrix under thermal environment. Based on the model, an explicit formula for the critical axial stress is derived in terms of the buckling modes of the shell and the parameters that indicate the effects of temperature change, surrounding elastic medium and the van der Waals forces. Based on that, some simplified analysis is carried out to estimate the critical axial stress for axially compressed buckling of the double-walled carbon nanotube. Numerical results for the general case are obtained for the thermal effect on axially compressed buckling of a double-walled carbon nanotube. It is shown that the axial buckling load of double-walled carbon nanotube under thermal loads is dependent on the wave number of axially buckling modes. And a conclusion is drawn that at low and room temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube increase as the value of temperature change increases, while at high temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube decrease as the value of temperature change increases.     

Thermoelastic buckling of steel columns with load-dependent supports

The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient α or the stiffening rate β of thermal restraints will increase the critical temperature.     

THERMOELASTIC VIBRATION OF FLUID-CONVEYING MICROTUBES EMBEDDED IN ELASTIC MEDIUMS

研究热环境中被弹性介质包围的微米输流管道的横向振动问题. 根据Hamilton 原理及非线性热弹性理论建立管道横向振动控制方程,并利用复模态法对其进行求解,得到了系统的固有频率和屈曲失稳临界流速,讨论了环境温度和一些重要系统参数对管道振动特性的影响. 研究结果表明:环境温度变化、管道和流体的微尺度效应、管道外径及弹性介质刚度对输流微管道固有频率和临界流速都有很大影响.     

Dynamic Stability of a Fluid-Coveying Cantilevered Pipe on an Additional Combined Support

Based on an analytical study, a numerical analysis is made of the dynamic stability of a cantilevered steel pipe conveying a fluid. The pipe is modeled by a beam restrained at the left end and supported by a special device (a rotational elastic restraint plus a Q-apparatus) at the right end. The numerical analysis reveals that the critical velocity of the fluid depends on the governing parameters of the problem such as the ratio of the fluid mass to the pipe mass per unit length and the rotational elastic constant at the right end     

Torsional buckling of a double-walled carbon nanotube embedded in an elastic medium

The torsional buckling of a double-walled carbon nanotube embedded in an elastic medium is studied in this paper. The effects of surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model is presented for the torsional buckling of a double-walled carbon nanotube. Based on the model, a condition is derived in terms of the buckling modes of the shell and the parameters describing the effect of van der Waals interaction and surrounding elastic medium. A simplified analysis is also carried out estimate the critical torque for torsional buckling of the double-walled carbon nanotube.     

Stability of an elastic cytoskeletal tensegrity model

An elastic cytoskeletal tensegrity structure composed by six inextensible elastic struts and 24 elastic cables is considered. The model is studied, adopting delay convention for stability. Critical conditions for simple and compound instabilities are defined. Post-critical behavior is also described. Equilibrium states with buckling of the struts are also considered. It is revealed that critical Euler buckling load of the struts is a necessary but not a sufficient condition for the existence of bifurcated equilibrium states, caused by buckling of the struts.