Crack effect on dynamic stability of beams under conservative and nonconservative forces

The purpose of this paper is to investigate the dynamic stability of beams containing a single crack subjected to conservative and nonconservative forces.The governing equation of the system can be derived from the extended Hamilton’s principle in which the kinetic energy, the elastic potential energy, the conservative work and the nonconservative work must be taken into account.The local flexibility matrix of a beam of a rectangular cross-section with a single edge crack is employed in order to perform numerical analysis.The investigated cracked beams are subjected to triangularly distributed subtangential forces, which are the combination of axial and tangential forces.The studied cracked beams become unstable in the form of either flutter or divergence, depending on crack parameters and on the degree of nonconservativeness of the load, when boundary conditions are fixed.     

Dynamic instability of spinning launch vehicles modeled as thin-walled composite beams

This paper presents the results of the dynamic stability analysis of a flexible spinning launch vehicle subjected to thrust modeled as a thin-walled composite beam with circular cross section. Due to the presence of gyroscopic forces, we mainly aimed to find divergence and/or flutter instabilities and establish the stability boundaries of the spinning beam. For this purpose, we implemented a circumferentially uniform stiffness (CUS) layup configuration to exhibit the coupled motion of bending–bending–shear. The solution of the eigenvalue problem is handled by the extended Galerkin method (EGM), and we computed the results addressing the effects of various parameters such as spin speed, axial load, ply angle, aspect ratio and transverse shear on the dynamic stability of the beam. Insights revealed by this study contribute to the design of advanced aerospace structures modeled as thin-walled composite beams reflecting the potential influence of transverse shear and aspect ratio on dynamic stability characteristics. A notable contribution is that we show that divergence/flutter instabilities can be delayed or even avoided using the directionality property of composite materials.     

Surface effect on dynamic stability of microcantilevers on an elastic foundation under a subtangential follower force

For micro/nano structures, surface elasticity, surface stress and surface mass strongly affect mechanical behaviors of 1D beam-columns. This article studies dynamic stability of microcantilevers on an elastic foundation or embedded in an elastic matrix when subjected to a subtangential follower force, where the surface effects are taken into account. An exact characteristic equation is derived for clamped–free end supports. For differential tangency coefficients, the force–frequency interaction diagram is displayed and the influences of surface elasticity modulus, residual surface tension, surface mass and the elastic foundation are analyzed for conservative and non-conservative compressive forces. When the tangency coefficient vanishes, a cantilever column subjected to a conservative tip force is reduced, and conventional Euler buckling for a compressive axial load is recovered. When the tangency coefficient does not vanish, a generalized Beck’s column with the surface effects is tackled. When the tangency coefficient exceeds certain critical value, flutter instability take places. For a fixed frequency, the critical divergency and flutter loads as a function of the tangency coefficient are given for various surface influences from residual surface tension, surface elasticity, surface mass and the stiffness of the elastic foundation. The boundary map of stability, divergence and flutter domain is shown.     

Calculation of flutter and dynamic behavior of advanced composite swept wings with tapered cross section in unsteady incompressible flow

In this study, the aeroelastic stability and response of a swept composite wing in subsonic incompressible flow are investigated. The wing is modeled as an anisotropic tapered thin-walled beam with the circumferentially asymmetric stiffness structural configuration to establish proper coupling between bending and torsion. The structural model considers a number of nonclassical effects, such as transverse shear, material anisotropy, warping inhibition, nonuniform torsion, rotary inertia and three-dimensional strain effects. The aerodynamic strip method based on two-dimensional Wagner function in unsteady incompressible flow is used. Following the analysis, the mass, stiffness and the damping matrices of the nonconservative aeroelastic system are formed such that the extended Galerkin method and the separation of variables method can be employed. As a result, the coupled and linear governing system of dynamic equations is obtained. Then, by transforming matrices into the state-space and state-vector forms, the problem under study is finally converted into an eigenvalue problem. The flutter and the divergence speeds for various layer configurations with different geometric and material properties and fiber orientations are obtained. By solving the aforementioned equations of motion in the time domain, the aeroelastic responses of the tapered swept composite wing are computed. The obtained results are compared with the available literature.     

Efficient laminated composite beam element subjected to variable axial force for coupled stability analysis

A numerically efficient laminated composite beam element subjected to a variable axial force is presented for a coupled stability analysis. The analytical technique is used to present the thin-walled laminated composite beam theory considering the transverse shear and the restrained warping-induced shear deformation based on an orthogonal Cartesian coordinate system. The elastic strain energy and the potential energy due to the variable axial force are introduced. The equilibrium equations are derived from the energy principle, and explicit expressions for the displacement parameters are presented using the power series expansions of displacement components. Finally, the member stiffness matrix is determined using the force–displacement relations. In order to verify accuracy and efficiency of the beam element developed in this study, numerical results are presented and compared with results from other researchers and the finite beam element results, and the detailed finite shell element analysis results using ABAQUS; especially, the influence of variable axial forces, the fiber orientation, and boundary conditions on the buckling behavior of the laminated composite beams is parametrically investigated.     

含分层损伤复合材料加筋层合板的动承载能力

采用有限元方法研究了含穿透分层损伤复合材料加筋层合板的动力响应和承载能力。根据复合材料层合板一阶剪切理论, 推导了复合材料层合板单元的刚度阵和质量阵列式;同时采用Adams 应变能法与Rayleigh阻尼模型相结合的方法, 构造了相应的阻尼阵列式;为了防止在低阶模态中分层处出现的上、下子板不合理的嵌入现象, 建立了含分层损伤复合材料加筋层合板动力分析中分层分析模型和虚拟界面联接模型。并采用Tsai提出的刚度退化准则和动力响应分析的精细积分法, 对在动荷载作用下含分层损伤复合材料加筋层合板结构进行了破坏和承载能力分析。通过典型算例分析, 分别讨论了外载频率、分层深度、筋的位置以及破坏过程中刚度退化对含损伤复合材料加筋层合板动力响应特征和承载能力的影响, 得到了一些具有理论和工程价值的结论。     

Post-critical behaviour of a uniform cantilever column subjected to a tip concentrated subtangential follower force with small damping

Large deflection analysis of a uniform cantilever column under a tip concentrated subtangential follower force has been studied using the dynamic criterion. The problem is solved with a simple and reliable iterative numerical scheme. The eigencurve, post-critical load and corresponding coalescence frequency parameter wherever applicable for different values of the subtangential parameter and the tip angle of the column are presented. The post-critical loads are also determined by considering arbitrarily (vanishingly) small internal damping forces. It is noted that small damping has a destabilizing effect.     

Stability of a column with a follower load and a load-dependent elastic support

Summary The influence of an elastic support on the stability of an elastic column under a follower load is investigated. The support is pinned and has a rotational spring whose stiffness is either constant or increases as the load is applied. Linear and quadratic stiffening functions are treated. An adjoint variational principle and a generalized Rayleigh-Ritz method are developed for this system. Characteristic curves are examined and the critical load, which is associated with flutter instability, is determined. The effects of the initial spring stiffness and the stiffening rate on the critical load are studied. It is found that an increase in either one of these parameters may lower the critical load, which is unexpected, and that support stiffening may have a significant influence on the vibrations and stability of a nonconservative system.     

Eigen analysis of fiber-reinforced composite plates

Composites plates of various orientation and thickness with fiber-reinforced plastics (FRP) find increasing applications in aerospace and automotive structures (due to their high strength, stiffness to weight ratios, and high damping characteristics). Determination of the dynamic characteristics and the investigation of such structures are essential not only in the design but also in manufacture development.

In the present paper, the eigen analysis of laminated square plates with various fiber orientations, various boundary fixations, and different stacking sequences has been presented and discussed. The experimental analysis and finite element techniques are utilized to study the effect of fiber orientation and boundary conditions on the dynamic characteristics of frequency and mode shapes. The vibrational system technique is utilized for experimental measurements. The results show agreement between experimental and theoretical investigations. Also, these results show a close connection between damping and stiffness characteristics in glass-fiber composites.     

Dynamic behavior of cross-ply laminated beams with piezoelectric layers

The dynamic behavior of cross-ply non-symmetric composite beams, having uniform piezoelectric layers is analysed. A first-order Timoshenko type analysis is applied to obtain the equations of motion, which include shear deformation, rotary inertia, bending-stretching coupling terms and induced axial strains caused by the piezoelectric material. Using the principle of virtual work, the coupled equations of motion and the relevant boundary conditions are obtained. For a laminated beam having uniform piezoelectric layers the induced strains appear only in the boundary conditions yielding time dependent ones. Therefore, a special procedure involving orthogonality of the coupled Timoshenko type natural vibrational modes of the beam is applied to help understanding of the dynamic behavior of the non-symmetric laminated beam and to investigate the influence of the induced strains (by the piezoelectric layers) on the dynamic behavior while keeping an ‘open-loop’ control. Typical types of laminates and piezoelectric materials are used to calculate natural frequencies and mode shapes. Numerical results for various parameters of laminated beams are presented to stress the better applicability and suitability of the present approach to the analysis of dynamic behavior of laminated composite beams with piezoelectric layers.     

Buckling and postbuckling behavior of delaminated sandwich beams

An investigation was performed to study the buckling and postbuckling behavior of sandwich beams containing lengthwise and depthwise through-the-width delaminations. An analytical beam model was developed to predict the buckling load of the beam and to describe its postbuckling response for arbitrarily situated delaminations and various combinations of boundary conditions. Griffith's energy release rate model was employed to predict the stability of delamination propagation under external loading and to determine the direction of delamination growth.

Parametric studies over a wide range of beam geometries, damage sizes and locations, composite facings and beam boundary conditions were carried out to study their effects on the overall behavior of the sandwich structure, as well as its damage tolerance. The results demonstrated that a sandwich construction is very ‘sensitive’ to the presence of delaminations situated at the core-faceplate interface. Premature buckling failure occurs at external loads which are significantly lower than the buckling load for a ‘perfect’ sandwich beam; in ‘imperfect’ beams with composite faceplates, the layup sequence affects significantly the load-carrying capacity of the beam; varying either the boundary conditions in a sandwich beam or the lengthwise location of a delamination has a small effect on the postbuckling behavior of the beam. Delaminations located within composite faceplates have less pronounced influence, and as the defect is moved outwards the limit load may reach the buckling load corresponding to that of the ‘perfect’ beam.

The proposed model is capable of analyzing the postbuckling behavior of both sandwich and composite laminated beams for arbitrary locations of the delamination, and various combinations of boundary conditions.     

桥梁颤振概率性评价的随机有限元法

在桥梁的各种风振形式当中,颤振对桥梁的安全威胁最大,而在各种随机因素作用下的桥梁颤振概率性评价也受到了人们的重视。为了计算大跨度桥梁的颤振失效概率,推出了一种结合有限元分析和可靠度计算理论的随机有限元方法,考虑了质量、刚度、阻尼和颤振导数等随机因素对颤振临界风速的影响,并对我国第一座真正意义上的大跨度海上桥梁——东海大桥颗珠山斜拉桥进行了颤振可靠性分析。     

GDQR求解弹性地基上输流管道的稳定性

用广义微分求积法（GDQR）研究了弹性地基上输流管道的稳定性问题。基于输流管道运动微分方程及边界条件,采用GDQR进行离散化,获得由动力方程组及边界条件合成的特征值矩阵方程。通过对相应特征值方程的具体分析,计算了左端固定、右端弹性支承下输流管道的发散失稳流速和颤振失稳流速,研究了临界失稳流速和稳定区域随两端支撑弹簧刚度、扭转弹簧刚度的变化情况,分析了质量比、双参数模型地基反力系数和剪切模量对输流管道稳定区域图的影响,得到了一些有益的结论。研究结论对于工程实践有一定的指导意义。     

Flutter of rectangular composite plates with variable fiber pacing

The effect of variable fiber spacing on the supersonic flutter of rectangular composite plates has been investigated using the finite element method and quasi-steady aerodynamic theory. The formulation of the location dependent stiffness and mass matrices due to nonhomogeneous material properties was derived. This study first demonstrates the flutter analysis of composite plates with variable fiber spacing. Numerical results show that the sequence of the natural mode may be altered and the two natural frequencies may be close to each other due to the fiber distribution may change the distributed stiffness and mass of the plate. Therefore, it may change the flutter coalescent modes. It is seen that the flutter boundary may be increased or decreased due to variable fiber spacing. The punch phenomenon on the flutter boundary is obvious in the absence of aerodynamic damping.     

基于几何因子的复合材料层合板颤振特性

首先实现了基于几何因子的复合材料层合板建模方法,解决了几何因子与Natran的参数输入问题,并通过一个简单算例进行验证。其次,在基于几何因子的层合板建模方法的基础上,采用p-k法计算颤振速度和发散速度,进行基于几何因子的悬臂复合材料层合板颤振和发散特性分析研究,重点研究了主轴刚度和弯扭耦合效应对颤振速度的影响。分析结果表明:相对于弯曲刚度,扭转刚度的改变对颤振速度的影响更显著,且扭转刚度越小,颤振速度越低;颤振模式随着刚度特性的改变有可能发生转变,导致颤振速度的突然变化和几何因子空间内颤振速度等高线的不连续;在正则化刚度矩阵不变的情况下,层合板厚度增加会同时提高颤振速度和发散速度,且颤振速度与发散速度与厚度大致呈线性关系。     

Nonlinear vibration and postbuckling of unsymmetrically laminated imperfect shallow cylindrical panels with mixed boundary conditions resting on elastic foundation

This study is analytically concerned with the titled problem. The rotational stiffness variation is assumed to be identical along opposite edges. The panel is subjected to inplane edge forces but not tangential boundary forces. A unified approximate solution is formulated on the basis of the dynamic Marguerre-type equations. The edge condition for the rotational stiffness variation is satisfied by expansion of the edge bending moments and the varying rotational edgerestraint coefficients into generalized Fourier series. These moments are also replaced by an equivalent lateral pressure near these edges. The Galerkin procedure furnishes an equation for the time function which is solved by the method of perturbation. In the postbuckling case the equation reduces to a relation between the postbuckling load and the maximum deflection. Numerical results for nonlinear vibration and postbuckling behavior of orthotropic and unsymmetrically laminated angle-ply cylindrical panels are presented graphically for different parameters and compared with available data.     

Aerothermoelastic analysis of composite laminated plates

The flutter behavior of a thermally buckled composite laminated plate is investigated in the frequency and time domains using the finite element method. Von Karman large deformation assumptions and quasi-steady aerodynamic theory are employed for the analysis. The effects of temperature gradient, panel length-to-width ratio, fiber orientation, and stacking sequence on aerothermoelastic behavior of the plate are studied in detail. The flutter boundary, buckling boundary, time history response, and phase plane plots of cross-ply and angle-ply laminates are presented. The numerical results show that temperature gradient induces thermal moments and increases the overall stiffness of the plate, and thus may increase the flutter boundary significantly. When the buckle pattern of the plate changes, the eigenvalues of the natural modes are changed suddenly and the sequence of the natural modes may be altered. Therefore, the change in the buckle pattern postpones the coalescence and the flutter boundary may increase. The change in the coalescence pair may also postpone the coalescence and increase the flutter boundary.     

Thermal dynamic stability of parametrically excited laminated composite plates with temperature-dependent properties

Thermal dynamic stability analysis is performed on periodically-loaded laminated composite plates with temperature-dependent properties. The periodic load is taken to be a combination of periodic axial and bending stress. A set of differential equations of Mathieu–Hill type is formed to determine the dynamic instability regions based on Bolotin's method. The thermal dynamic instability of laminate plates with respective temperature-dependent and temperature-independent properties is examined. The effects of various parameters on the instability region and dynamic instability index are also discussed. The results show that the temperature-dependent properties have a significant influence on the thermal dynamic behavior of laminated plates.     

边界约束对梁抗爆动力响应的影响(Ⅰ)–理论研究及分析

边界约束的差异会直接影响结构的抗爆动力响应及承载能力,文中建立了复杂约束条件下抗爆梁在弹性阶段和塑性阶段的解析计算方法,并计算分析了竖向弹性与阻尼约束、水平约束刚度、抗弯约束、荷载形式以及屈服弯矩动力强化系数对动力响应的影响。计算表明：竖向弹性与阻尼约束会引起附加惯性力,能够明显降低结构在弹塑性阶段的位移动力系数。水平约束和抗弯约束影响结构的动态响应主要在塑性阶段,水平约束使梁截面在变形过程中产生横向压力,抗弯约束直接限制刚体转动,均有效降低了梁位移动力系数,相对提高结构的承载力。相同约束刚度和荷载峰值条件下,平台荷载下结构的位移动力函数均高于三角形荷载下位移动力函数,说明动荷载的作用时间越长,对结构承载越不利。另外考虑屈服弯矩的动力增强系数时,可提高结构的抗爆潜力。     

弹性地基上受切向力作用梁稳定性研究

用广义微分求积法(GDQR)分析了弹性地基上复杂弹性支承条件下受切向力作用梁的稳定性问题。基于弹性支承梁的运动微分方程及边界条件,采用GDQR进行离散化,获得由动力方程组及边界条件合成的特征值矩阵方程。通过对相应特征值方程的具体分析,讨论了弹性地基模量、剪切系数、复杂边界条件对临界载荷的影响,研究了一端固定约束、另一端弹性约束梁弹性失稳区域随弹性地基模量和支承弹簧刚度变化的情况,得到了一些有益的结论。结果表明：GDQR能很好地解决此类系统的稳定性问题。