Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.     

Analysis of the Effect of Changed Curvature and Load Distribution on the Deformation of a Flexible Long Cylindrical Shell

The effect of changed curvature and load distribution on the deflection–load relationship is studied based on the exact analytical solutions of the nonlinear problem on the deformation of a flexible long noncircular cylindrical shell with clamped and hinged edges under a nonuniform normal load. Graphs show how changes in the curvature and load parameters affect the upper critical load.     

Refined theories may be needed for vibration analysis of structures with overhang

The effect of shear deformation and rotary inertia terms on the free vibration of a beam with overhang was investigated. A recently proposed modified Timoshenko-type equations of motion were used to analyze the vibration of the structure. Two different sets of boundary conditions, with either a fixed or hinged end support, were studied. The results were compared with those obtained for the classical Bernoulli–Euler beam theory. The comparison shows that for a hinged end beam with very long overhang, where the span between the supports is less than one tenth of the overall beam length, the classical theory significantly overestimates the values of the fundamental natural frequencies, even for isotropic shear rigidity. On the other hand, the span effect is reversed for the clamped end beam, for which a relatively significant difference between the classical theory and shear theory results may occur only for a long span. For transversely isotropic beams, the refined theory predictions of the fundamental natural frequencies may be much smaller than those obtained through the rigid shear theory, especially for short span hinged end beams and long span clamped end beams.     

Stability of a straight bar of variable rigidity

We consider the longitudinal compression of a straight bar whose rigidity is a periodic integrable function of the longitudinal coordinate. For a hinged bar with one clamped end, we obtain approximate analytic formulas that permit obtaining the critical compressing loads under which an adjacent, curved form of equilibrium is possible. In the case of a bar of stepwise varying rigidity that consists of a single period (the limit case), we compare the results obtained by our formulas with the already known exact solutions of the stability equation. A good agreement between the approximate and exact results is shown.     

Buckling of cantilever cones under external pressure

An experimental and numerical study was performed on the buckling behavior of clamped-free truncated cones under external pressure. A lap-jointed polyester cone was vertically settled in a water vessel and was clamped at the small radius end. Raising water level, the relation of normal displacement with water level was recorded and the buckling water level was determined. The experimental results were in good agreement with the finite-element solutions. The same finite-element program was then used for a parametric study on the buckling of clamped-free cones subjected to uniform external pressure. The results show that the ratio of the critical external pressure of a clamped-free cone with that of a simply supported equivalent cylinder is a function of only the taper ratio ψ of the cone.     

Two-dimensional elastica analysis of equilibrium shapes of single-anchor inflatable dams

Several types of inflatable dams are considered. These are long, air-inflated, cylindrical structures on a rigid foundation. Sometimes one of the long edges of a sheet is folded back to the other edge, and then the two edges are clamped to the foundation along a single anchoring line. A second configuration can be modeled as two sheets attached along two long edges, with one edge anchored and the other free to lift as air pressure is applied between the sheets. Another device treated here is a hinged spillway gate lifted by an inflatable bladder. The cross section of the dam or bladder is analyzed as an inextensible elastica. The governing equations and boundary conditions are formulated for each case, and shooting methods are utilized to obtain numerical solutions for the equilibrium shapes. The effects of the internal air pressure and the external water height are investigated.     

子弹冲击下TiNi固支梁的结构动态响应特性研究

采用改装的霍普金森压杆装置结合数值模拟对伪弹性TiNi合金固支梁的结构动态响应特性进行了研究。结果表明,在子弹冲击下,撞击点和固定端附近首先发生相变,并随着载荷增加,进一步产生相变铰,梁演变为二杆铰接机构。由于轴力作用,此处相变铰为拉伸侧的单边铰。与传统塑性铰不同,卸载后相变铰完全消失,梁回复原状没有残余变形。此外,对固支边界条件的实现及其对实验结果的影响进行了专门研究。     

Dynamic Behavior of Rigid–Plastic Sector Plates

The paper studies the dynamic behavior of perfect rigid-plastic plates in the form of a sector with hinged or clamped sides under short-term intensive loads. Two dynamic deformation mechanisms are demonstrated. The dynamic equation is derived for each of the mechanisms. The realization conditions for the mechanisms are analyzed. Analytical expressions are derived for the ultimate (high) loads and the maximum residual deflection. Numerical examples are given     

近场水下爆炸气泡脉动及水射流的实验与数值模拟研究

海上作战时,近场水下爆炸形成的水射流能造成水面舰船结构的严重局部毁伤。为了研究近场爆炸时舰船底部水射流的形成机理及规律,开展了TNT当量2.5 g的炸药在固支方板底部不同爆距下起爆的水下爆炸实验。结果表明,气泡坍塌形成水射流的过程随着爆距的增加由吸附式向非吸附式转化。接着,基于ABAQUS软件采用CEL方法开展了系列数值模拟,结果表明:爆距在0.821～0.867倍最大气泡半径时,存在吸附式射流向非吸附式射流转化的临界点;固支方板加快了气泡坍塌的进程,炸药与钢板间的距离越小则射流形成的时间越早;射流形成过程中最大速度和射流击中钢板时速度均随着爆距的增大先增大后减小,并在临界点附近达到最大值,射流速度最大可达621 m/s,射流击中钢板时速度最大可达269 m/s。最后,给出了射流开始形成时间、射流最大速度、射流最大速度出现时间、射流击中钢板速度和射流击中钢板时间与距离参数的函数关系式。     

On the instabilities of an externally loaded shallow spherical shell1

The simple von Kármán model of a clamped shallow elastic cap subjected to external pressure is reformulated as an elementary catastrophe. Conceptual understanding of load deflection behavior is substantially improved as a result. Three distinct modes of deflection behavior are identified. One snap-through type behavior is substabtiated by comparison to experimental data.     

ADHESIVE CONTACT PROBLEM OF AXISYMMETRIC MINIATURE CIRCULAR PLATES WITH CENTRAL RIGID BUMP

Considering the adhesive effect and geometric nonlinearity, the adhesive contactbetween an elastic substrate and a clamped miniature circular plate with two different centralrigid bumps under the action of uniform transverse pressure and in-plane tensile force in theradial direction was analyzed. And an analytical solution is presented by using the perturbationmethod. The relation of surface adhesive energies with critical load to detach the contacted surfacesis obtained. In the numerical results, the effects of adhesive energy, in-plane tensile force, rigidbump size and contact radius on the critical load are discussed, and the relation of critical contactradius with the gap between the central rigid bump and the substrate for different adhesive energiesis investigated.     

On the nonlinear stability of symmetrically laminated cylindrically orthotropic shallow spherical shells

In this paper,the large deflection theory of symmetrically laminated cylindricallyorthotropic shallow spherical shells is established.Based on this theory,applying themodified iteration method,the analytic solution for critical buckling loads of the shells withrigidly clamped edges under actions of uniform pressure has been obtained.     

Behavior of rigid-plastic polygonal plates with a rigid insert under blast loads

A method based on a perfect rigid-plastic body model is developed to analyze the dynamic behavior of hinged or clamped polygonal plates that have a perfectly rigid insert and rest on a viscoelastic foundation with supports. The plate is subject to an arbitrary blast load of high intensity uniformly distributed over the plate surface. Two cases of plate deformation are examined. In each of the cases, equations of motion are derived and realization conditions are analyzed. Analytic expressions for the deformation time and the maximum residual deflection are derived in the case of an arbitrary load of medium intensity and in the case of high-intensity load described by a rectangular function. Examples of numerical solutions are given __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 99–110, January 2008.     

Buckling of cylindrical shells heated along two opposite generators combined with axial compression

A series of tests was carried out on clamped cylindrical shells under a combination of thermal and mechanical loads. The shells were linearly heated along two opposite generators in addition to a uniform axial pressure applied before the heating. The paper describes the test setup, equipment and techniques used. Systems for rapid measurement, data analysis and storage designed for the experimental work are presented. As an outcome of this study, a linear-interaction line is proposed in order to express the interaction between thermal-and mechanical-buckling loads.     

Numerical analysis of axisymmetric buckling of conical shells

Nonlinear boundaryvalue problems of axisymmetric buckling of conical shells under a uniformly distributed normal pressure are solved by the shooting method. The problems are formulated for a system of six firstorder ordinary differential equations with independent rotation and displacement fields. Simply supported and clamped cases are considered. Branching solutions of the boundaryvalue problems are studied for different pressures and geometrical parameters of the shells. The nonmonotonic and discontinuous curves of equilibrium states obtained show that collapse, i.e., snapthrough instability is possible. For a simply supported shell, multivalued solutions are obtained for both external and internal pressure. For a clamped thinwalled shell, theoretical results are compared with experimental data.     

均布压力下正交异性圆锥扁壳的后屈曲分析

基于Ｋａｒｍａｎ型大挠度方程,用修正迭代法分析了均布压力下夹支正交异性圆锥扁壳的几何非线性的后屈曲行为,给出二阶近似的荷载挠度特征关系式及临界荷载,给出了三种正交异性参数对应数值结果,分析了正交异性参数对壳体变形和屈曲荷载的影响。     

固—液耦合Timoshenko管道的稳定性分析

根据Ｈａｍｉｌｔｏｎ原理的固－液耦合振同分方程用幂级数法计算了Ｔｉｍｏｓｈｅｎｋｏ管道的固有频率和临界流速。给出了管道前三阶固有频率－流速的关系曲线,分析了转动惯量对该输流管道的稳定特性的影响。计算结果表明,转动一对两端简支的固－液合Ｔｉｍｏｓｈｅｎｋｏ管道的静力失稳没有影响,但对其频率特性和动力失稳有影响。     

Experimental study on a two-phase critical flow with a non-condensable gas at high pressure conditions

An experimental study was performed on a two-phase critical flow with a non-condensable gas at high pressure conditions. Experimental data for the critical flow rates were generated by using sharp-edged stainless steel pipes with an inner diameter of 10.9 mm, a thickness of 3.2 mm, and a length of 1000 mm. The test conditions were varied by using the stagnation pressures of 4.0, 7.0, and 10.0 MPa, water subcoolings of 0.0, 20.0, and 50.0 °C, and nitrogen gas flow rates of 0.0–0.22 kg/s. The experimental results show that the critical mass flux decreases rapidly with an increase of the volumetric non-condensable gas fraction. Also the critical mass flux increases with an increase of the stagnation pressure and a decrease of the stagnation temperature. An empirical correlation of the non-dimensional critical mass flux, which is expressed as an exponential function of the non-condensable gas fraction of the volumetric flow, is obtained from the experimental data.     

The singular perturbation method applied to the nonlinear stability problem of a shallow spherical shell (II)

In this paper we consider the nonlinear stability of a thin elastic circular shallow spherical shell under the action of uniform normal pressure with a clamped edge. When the geometrical parameter k is large, the uniformly valid asymptotic solutions are obtained by means of the singular perturbation method. In addition, we give the analytic formula for determining the centre deflection and the critical load, and the stability curve is also derived. This paper is a continuation of the author’s previous paper[11].     

Collision, damage, smooth evolution of an articulation. The “tennis elbow”

We present a simple example of phenomenon where collision and damage take place together. We consider an arm with its forearm connected by the elbow: a plane system made of two rods, one of which is clamped and the other is hinged at its extremum and free to rotate about it. The rotation angle ? of the forearm is constrained to be neither larger than π nor lower than 0. The rods are connected through a hair spring, which represents the elbow articulation and whose damage is accounted for. The damage may be related to the tennis elbow pathology. We also take into account the discontinuity of the angular velocity of the forearm, due to collisions when ?=0 or ?=π, and provide some numerical results.