Analysis of Stresses and Displacements in Orthotropic Noncircular Cylindrical Shells Under Centrifugal Loads

This paper addresses the class of stress–strain problems for thin orthotropic cylindrical shells of arbitrary cross section under centrifugal loads. Separating out the variables for a simply supported shell yields a system of ordinary differential equations for which the boundary-value problem is solved by a stable numerical method. Study is made of the distribution of displacements in shells of elliptical cross section versus the ratio of ellipse exes and the eccentricity of the axis of revolution relative to the geometrical axis of symmetry     

The null-field approach and the propagation of elastic surface waves along cylindrical cavities

The existence of surface wave modes, propagating along an infinite cylindrical cavity of general, constant cross section is demonstrated for two non-elliptic geometries. The method used is the null-field approach with global expansions of the surface displacement. The use of Fourier expansions for the surface fields is carefully investigated. Numerical results are presented for the phase velocities and the surface displacements.     

单层偏心圆柱薄壳的自由振动特性分析研究

本文从偏心圆柱壳截面的几何特性出发,将偏心圆柱壳问题转化为一个周向变厚度圆柱壳问题,随后利用其状态向量之间的传递矩阵将壳体的振动控制方程转化为矩阵微分方程形式,通过Magnus级数法求解传递矩阵得到频率方程。采用Lagrange插值法得到偏心圆柱壳体自由振动状态下的固有频率,并且与圆柱壳的固有频率进行了比较。对影响结构固有频率的主要参数进行了分析,得到了这些参数和固有频率之间的关系。本文不仅提出了一种有效求解偏心圆柱壳固有频率的新方法,同时亦可为检测偏心圆柱壳的偏心距提供一种新的思路和方法。     

Formulation of Problems in the Bernoulli—Euler Theory of Anisotropic Inhomogeneous Beams

A procedure of reducing the three-dimensional problem of elasticity theory for a rectilinear beam made of an anisotropic iuhomogeueous material to a one-dimensional problem on the beam axis is studied. The beam is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the beam from its end to any cross section. The internal force factors are related to the characteristics of the strained axis under the prior assumptions on the distribution of displacements over the cross section of the beam. To regulate these assumptions, the displacements of the beam’s points are expanded in two-dimensional Taylor series with respect to the transverse coordinates. Some physical hypotheses on the behavior of the cross section under deformation are used. The well-known hypotheses of Bernoulli—Euler, Timoslienko, and Reissner are considered in detail. A closed system of equations is proposed for the theory of anisotropic iuhomogeueous beams on the basis of the Bernoulli—Euler hypothesis. The boundary conditions are formulated from the Lagrange variational principle. A number of particular cases are discussed.     

斜拉桥分析的索膜单元理论及其应用

本文提出分析斜拉桥的索膜单元理论,建立起索膜单元概念与计算模型,推导出索膜单元刚度矩阵,说明了用索膜单元分析斜拉桥的方法细节,应用索膜单元对144+360+144米公路钢斜拉桥方案进行了分析计算,得出了主梁弯矩影响面与扭矩图,讨论了索梁锚固点与截面水平主惯性轴偏心对主梁弯矩的影响。     

Transversal oscillations of a fluid in a long cylindrical container with membrane or elastic plate on the free surface

We propose approximate solutions of two-dimensional hydroelastic problems that describe free oscillations of an ideal fluid in a horizontal long cylindrical container with arbitrary symmetric cross section. The free surface of the fluid is covered by a plane membrane or an elastic plate. Using specific examples, we analyze the obtained solutions and the results of computation of frequencies and forms of oscillations of the mechanical system under consideration.     

Nonlinear analysis of beams of variable cross section,including shear deformation effect

In this paper the analog equation method (AEM), a BEM-based method, is employed for the nonlinear analysis of a Timoshenko beam with simply or multiply connected variable cross section undergoing large deflections under general boundary conditions. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of the shear deformation effect is remarkable.     

On the validity of planar, thick curved beam models derived with respect to centroidal and neutral axes

Most dynamic analyses of planar curved beams found in the literature are carried out based on a curved beam model which assumes that the neutral axis coincides with the centroidal axis of the curved beam. This assumption leads to governing equations of motion which are relatively simple with analysis results that have acceptable accuracy for shallow curved beams. However, when a curved beam is not shallow and/or its cross section is not doubly symmetric, the offset distance between the neutral and centroidal axes may be large enough to influence the in-plane dynamics of the curved beam even for small motion. In this paper, the validity of this underlying assumption for modeling a linear curved beam is examined. To this end, two sets of equations of motion governing the in-plane dynamics of a planar curved beam are derived, in a consistent manner for comparison, based on the linear strain-displacement relations and Hamilton’s principle. The first set of equations is derived from the displacement components measured with reference to the neutral axis of the curved beam while the second set is derived with respect to the centroidal axis of the cross section. The curved beam is considered extensional and the effects of rotary inertia and radial shear deformation are included. In addition to the curvature parameter that characterizes the wave motion for both curved beam models, an eccentricity parameter is introduced in the first model to account for the offset between the neutral and centroidal axes. The dynamic behavior predicted by each curved beam model is compared in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and modeshapes, and frequency responses. In order to ensure that the comparison is accurate, the wave propagation technique is applied to obtain exact wave solutions. It is shown that, when the curvature parameter is not small, the underlying assumption has a substantial impact on the accuracy of the linear dynamic analysis of a curved beam.     

Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with nonuniform cross section

Free vibration of nonuniform axially functionally graded Timoshenko beams subjected to combined axially tensile or compressive loading is studied. An emphasis is placed on the effect of tip and distributed axial loads on the natural frequencies and mode shapes for an inhomogeneous cantilever beam including material inhomogeneity and geometric non-uniform cross section. The initial value method is developed to determine the natural frequencies. The method’s effectiveness is verified by comparing our results with previous ones for special cases. Natural frequencies of standing/hanging Timoshenko beams are calculated for four different cross sections. The influences of shear rigidity, taper ratio, gradient index, tip force, and axially distributed loading on the natural frequencies of clamped-free beams are discussed. Material inhomogeneity and geometric non-uniform cross-section strongly affect higher-order vibration frequencies and mode shapes.     

Shear deformation effect in non-linear analysis of composite beams of variable cross section

In this paper the non-linear analysis of a composite Timoshenko beam with arbitrary variable cross section undergoing moderate large deflections under general boundary conditions is presented employing the analog equation method (AEM), a BEM-based method. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson's ratio and are firmly bonded together. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of non-linear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples are worked out to illustrate the efficiency, the accuracy, the range of applications of the developed method and the influence of the shear deformation effect.     

一种高阶非线性钢筋混凝土平板单元

在文献［１］高阶钢筋混凝土膜元的基础上，提出了一种具有１４个节点的高阶钢筋混凝土平板单元．钢筋混凝土材料模型仍采用Ｖｅｃｃｈｉｏ的抹平旋转裂缝模型，几何非线性仍采用所谓总体Ｌａｇｒａｎｇｅ列式法，非线性方程采用割线刚度位移增量迭代法．数值算例表明本文的方法是可靠的，高阶平板单元虽然列式复杂，但与低阶元相比，其计算量要少且精度要高     

Three-dimensional instability of an elliptic Kirchhoff vortex

The instability of a Kirchhoff vortex [1–3] with respect to three-dimensional perturbations is considered in the linear approximation. The method of successive approximations is applied in the form described in [4–6]. The eccentricity of the core is used as a small parameter. The analysis is restricted to the calculation of the first two approximations. It is shown that exponentially increasing perturbations of the same type as previously predicted and observed in rotating flows in vessels of elliptic cross section [4–9] appear even in the first approximation. As distinct from the case of plane perturbations [1-3], where there is a critical value of the core eccentricity separating the stable and unstable flow regimes, instability is predicted for arbitrarily small eccentricity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1988.     

冷弯薄壁槽形截面钢柱的优化设计

求解冷弯开口薄壁截面短柱的极限承载力是一个板组结构的大挠度弹塑性分析问题，板件之间相互作用和纵向面内加载方式对其局部屈曲后性能和稳定极限承载力有很大的影响。本文应用大挠度弹塑性有限条分析方法系统地研究了冷弯薄壁槽形截面柱的极限承载能力和它们的优化性能，并对荷载偏心距为常数和位移梯度为常数的两种加载条件下的薄壁开口短柱局部屈曲后性能和极限承载力进行了对比分析和探讨。     

Generation of harmonics in a waveguide with moving boundaries

A wave propagating through a waveguide with constant cross section has a finite number of propagating modes with the corresponding well-defined frequencies. When the boundaries of the waveguide vary with time while the cross section remains finite, higher modes are excited, and there will be an infinite number of frequencies associated with the propagating wave. The problem in this case is nonseparable, but an expansion in terms of partial waves yields an infinite set of coupled ordinary differential equations with time-dependent coefficients. It is shown that one can find the same set of equations by a series of canonical transformations applied to the Hamiltonian of the system. These transformations remove the time dependence of the boundaries in the Hamiltonian, but change the Hamiltonian density and make it explicitly time-dependent. The case of a rectangular waveguide with arbitrary motion of the boundaries is considered in detail as a typical example of such a waveguide.     

Dimensionally Scaled Miscible Displacements in Heterogeneous Permeable Media

Understanding and predicting the performance of solvent drives and remediation of contaminated aquifers in heterogeneous reservoirs is of great importance to the petroleum and environmental industries. In this paper, a general method to scale flow through heterogeneous reservoirs is presented for a miscible displacement of oil by a solvent. Results show that scaling miscible displacements in a two-dimensional, heterogeneous, anisotropic vertical cross section requires the matching of 13 dimensionless scaling groups. These groups were derived using a general procedure of inspectional analysis. A detailed numerical sensitivity study was performed to reveal the relationship between the scaling groups and the fractional oil recovery of miscible displacements in heterogeneous reservoirs. This relationship was then mapped using an artificial neural network, which can be used as a quick prediction tool for the fractional oil recovery for any combinations of the scaling groups, thus eliminating the need for the expensive fine-mesh simulations. These results have potential applications in modeling miscible displacements and in the scaling of laboratory displacements to field conditions.     

Nonlinear flexural–torsional dynamic analysis of beams of variable doubly symmetric cross section—application to wind turbine towers

In this paper, a boundary element solution is developed for the nonlinear flexural–torsional dynamic analysis of beams of arbitrary doubly symmetric variable cross section, undergoing moderate large displacements, and twisting rotations under general boundary conditions, taking into account the effect of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions and to twisting and/or axial loading. Four boundary-value problems are formulated with respect to the transverse displacements, to the axial displacement, and to the angle of twist and solved using the Analog Equation Method, a Boundary Element Method (BEM) based technique. Application of the boundary element technique yields a system of nonlinear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled with algebraic equations. Numerical examples of great practical interest including wind turbine towers are worked out, while the influence of the nonlinear effects to the response of beams of variable cross section is investigated.     

Solution of Boundary-Value Problems of the Dynamics of a Liquid in Horizontal Cylindrical Cavities with Partitions

We propose methods for the construction of approximate solutions of main boundary-value problems. These methods enable one to determine frequencies and attached masses of an ideal liquid in a moving horizontal cylinder that has an arbitrary symmetric cross section and structural elements in the form of longitudinal edges-partitions.     

On the behavior of viscoelastic free jets with elliptical cross section

In [1,2] we develop a comprehensive theory of one space dimensional closure models (closed systems of 1-D equations for the unknown, or modal, variables) for free viscoelastic jets. These closure models are derived via asymptotics from the full 3-D boundary value problem under the conditions of a Von Kármán-like flow geometry, a Maxwell-Jeffreys constitutive model, elliptical free surface cross section, and a slender jet scaling. The focus of the present paper is to determine the consequences and predictions of the lowest order system of equations in this asymptotic analysis. For the special cases of elliptical inviscid and Newtonian free jets, subject to the effects of surface tension and gravity, our model predicts oscillation of the major axis of the free surface elliptical cross section between perpendicular directions with distance down the jet, and draw-down of the cross section, in agreement with observed behavior. In the absence of surface tension the transformation from a cross section with major axis in one direction to a cross section with major axis in the perpendicular direction occurs only once, in agreement with the observation of Taylor [4]. In viscoelastic regimes, our model predicts swell of the elliptical extrudate and distortion of the elliptical extrudate cross section from the dimensions of the die aperture.     

含双裂纹圆截面悬臂梁的损伤识别研究

裂纹的萌生和扩展直接影响构件的振动响应,对构件的安全可靠性具有重要影响.本文以圆截面悬臂梁为对象,结合转角模态振型和模态频率等高线,研究了一种双裂纹识别技术.首先,基于应力强度因子和卡氏定理推导了无裂纹梁单元和含裂纹梁单元的刚度矩阵;在此基础上,建立了含裂纹圆截面悬臂梁的有限元动力学方程;然后,结合裂纹对梁转角模态振型和模态频率的影响,提出了双裂纹识别策略.最后,通过算例讨论了双裂纹识别策略的可行性.结果表明,圆截面悬臂梁的模态转角在裂纹位置出现突变,裂纹深度越大转角突变值越大;将识别出的裂纹位置作为已知参数,通过模态频率等高线法,可以准确地识别出双裂纹的深度.     

A quaternion-based formulation of Euler–Bernoulli beam without singularity

This paper proposes a singularity-free beam element with Euler–Bernoulli assumption, i.e., the cross section remains rigid and perpendicular to the tangent of the centerline during deformation. Each node of this two-nodal beam element has eight nodal coordinates, including three global positions and one normal strain to describe the rigid translation and flexible deformation of the centerline, respectively, four Euler parameters or quaternion to represent the attitude of cross section. Adopting quaternion instead of Eulerian angles as nodal variables avoids the traditionally encountered singularity problem. The rigid cross section assumption is automatically satisfied. To guarantee the perpendicularity of cross section to the deformed neutral axes, the position and orientation coordinates are coupled interpolated by a special method developed here. The proposed beam element allows arbitrary spatial rigid motion, and large bending, extension, and torsion deformation. The resulting governing equations include normalization constraint equations for each quaternion of the beam nodes, and can be directly solved by the available differential algebraic equation (DAE) solvers. Finally, several numerical examples are presented to verify the large deformation, natural frequencies and dynamic behavior of the proposed beam element.