Buckling analysis of a ring under the action of internal pressure in a cylindrical shell

Buckling analysis of a thin cylindrical shell stiffened by rings with T-shaped cross section under the action of uniform internal pressure in the shell is performed. An annular plate stiffened over the outer edge by a circular beam is used as the ring model. The classical ring model, which is a beam with a T-shaped cross section, is inappropriate in this problem, since in the case of the loss of stability, buckling deformations are localized on the ring surface. The beam model does not allow one to find the critical pressure that corresponds to such a loss of stability. In the first approximation, the problem of the loss of stability of the annular plate connected with the shell is reduced to solving the boundary value problem for finding eigenvalues of the annular plate bending equation. Approximate formulas for determining critical pressure are obtained under the assumption that the plate width is much smaller than its inner radius. The results found using the Rayleigh method and the shooting method differ slightly from each other. It has been demonstrated that the critical pressure for rings with rectangular cross section is higher than that for rings with a T-shaped cross section.     

Multilayer beams: A geometrically exact formulation

Summary We review and extend our recent work on a new theory of multilayer structures, with particular emphasis on sandwich beams/1-D plates. Both the formulation of the equations of motion in the general dynamic case and the computational formulation of the resulting nonlinear equations of equilibrium in the static case based on a Galerkin projection are presented. Finite rotations of the layer cross sections are allowed, with shear deformation accounted for in each layer. There is no restriction on the layer thickness; the number of layers can vary between one and three. The deformed profile of a beam cross section is continuous, piecewise linear, with a motion in 2-D space identical to that of a planar multibody system that consists of three rigid links connected by hinges. With the dynamics of this multi (rigid/flexible) body being referred directly to an inertial frame, the equations of motion are derived via the balance of (1) the rate of kinetic energy and the power of resultant contact (internal) forces/couples, and (2) the power of assigned (external) forces/couples. The present formulation offers a general method for analyzing the dynamic response of flexible multilayer structures undergoing large deformation and large overall motion. With the layersnot required to have equal length, the formulation permits the analysis of an important class of multilayer structures with ply drop-off. For sandwich structures, an approximated theory with infinitesimal relative outer-layer rotations superimposed onto finite core-layer rotation is deduced from the general nonlinear equations in a consistent manner. The classical linear theory of sandwich beams/1-D plates is recovered upon a consistent linearization. Using finite element basis functions in the Galerkin projection, we provide extensive numerical examples to verify the theoretical formulation and to illustrate its versatility. Dedicated to the memory of Professor Juan Carlos Simo, whose early demise is a great loss for the applied and computational mechanics community This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

Torsional vibrations and dispersion of torsional waves in orthotropic composite rods

The Barr’s refined theory of torsional vibrations of isotropic rods of noncircular cross section is generalized for an orthotropic material. An analysis of natural frequencies of torsional vibration of free-free orthotropic prismatic rods of rectangular cross section is carried out with the help of an exact solution of the frequency equation. For orthotropic CFRP and GFRP rods, the improved theory, which takes into account the normal stresses and inertia forces in the axial direction, in some cases, predicts a noticeable raise in the natural frequencies compared with those following from the Saint-Venant classical theory. A good agreement is obtained between the experimental and calculated values of natural frequencies of torsional vibrations of rectangular quartz and fiber glass rods. The dispersion of torsional waves in an orthotropic quasi-homogeneous rod is considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 165–182, March–April, 2008.     

Bending and torsion of anisotropic thin-walled open sections

A theory of thin-walled open sections, based on Vlasov's assumptions, is proposed. The section is assumed to be composed of rectangular strips whose cross section remains constant in the z-direction. The material is homogeneous and anisotropic with a single plane of elastic symmetry parallel to the middle surface of the strip. The general case of loading-tension, unsymmetrical bending, and constrained torision-is considered.Mekhanika Polimerov, Vol. 3, No. 5, pp. 900–905, 1967     

Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending

A mathematical model incorporating the higher order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. The rotor system considered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (deformation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed.     

夹芯梁的精确解法

夹芯梁与普通梁的本质区别在于剪切引起芯层横截面严重的而又不均匀的翘曲变形,其应力分布已远非初等理论所能描述,而正在广泛应用的经典夹层理论却都建立在平面假设基础上,尤其不能正确反映弱芯的轻质夹层结构的行为,本文放弃了不合理的假设,将夹芯梁视为一般层状弹性体,严格按弹性理论导出了既满足控制方程又同时满足全部边界条件、层间的应力及位移的连续条件的封闭解.它可确切地反映夹芯梁的位移形态和应力分布,并从不同角度,包括多种实验和FEM数值解,验证了它的正确性.     

The anisotropic coupling of one-dimensional problems in linear elasticity

We investigate the coupling of the four one-dimensional problems of linear elasticity (i.e., the rod-, the two beam- and the shaft-problem), for the case of a cross section that has two orthogonal axes of reflectional symmetry. An analytical proof is given, that the problems are decoupled for a homogeneous orthotropic material and the coupling behavior is given for monoclinic materials in dependence of the orientation of the symmetry plane. For a general anisotropic (aelotropic) material all four problems are coupled. We also identify the driving forces for the four problems, leading to a detailed definition of the admissible load-cases for the classical problems, so that any three-dimensional load-case can be uniquely decomposed into the driving forces of the four problems. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

叠层复合材料杆弯曲的层间应力(Ⅱ)

一矩形横截面的叠层复合材料杆,由以一种材料为中心部分,及另一种材料的上下两相同的盖板所组成.以线性弯曲应力所组成的力偶,作用于杆两端的中心部分(图1a),使杆弯曲.本文将探讨层间应力,以表明力是怎样通过胶合面传递给盖板的.     

Vibration analysis of Nano-Rotor's Blade applying Eringen nonlocal elasticity and generalized differential quadrature method

This study investigates the small scale effect on the flapwise bending vibrations of a rotating nanoplate. The nanoplate is modeled with a classical plate theory and considering cantilever and propped cantilever boundary conditions. Due to the rotation, the axial forces are included in the model as true spatial variation. Hamilton's principle is used to derive the governing equation and boundary conditions of the classical plate theory based on Eringen's nonlocal elasticity theory. The generalized differential quadrature method is employed to solve the governing equation. The effect of small-scale parameter, non-dimensional angular velocity, non-dimensional hub radius, aspect ratio, and different boundary conditions in the first four non-dimensional frequencies is discussed. Due to considering rotating effects, results of this study are applicable in nano-machines such as nano-motors and nano-turbines and other nanostructures.     

Mathematical model for the bending of plastically anisotropic beams

A mathematical model for the bending of a plastically anisotropic beam simply supported at both ends and subjected to a constant moment is considered. A differential equation with variable coefficients is derived for the beam curvature. The yield points of the beam material under tension and compression are assumed to be known. The elastoplastic bending of the beam with allowance for the strength-different (SD) effect is considered. The classical Bernoulli–Euler beam theory and the ideal plasticity model are used construct the mathematical model, and the problem is solved analytically. The solutions obtained for a classical isotropic beam and an SD beam are compared, and the contribution of the SD effect is analyzed. The problem is solved completely, and its results can be used to study bending under different loading.     

Closed-form solutions for stress singularities at bi-material wedges in Reissner-Mindlin plates

In this work, stress singularities in isotropic bi-material junctions are investigated using Reissner-Mindlin plate theory by means of a complex potential formalism. The governing system of partial differential equations is solved employing methods of asymptotic analysis. The resulting asymptotic near-fields including the singularity exponent λ are obtained in a closed-form analytical manner as solutions of a corresponding eigenvalue problem. The singular solution character is discussed for different geometrical configurations. In particular, the present study investigates the influence of the material constants on the singularity exponent. It is shown, that the Reissner-Mindlin theory allows for distinguishing between singularities of the bending moments and the transverse shear forces. Further, stronger singularities than the classical crack-tip singularity are observed. The results allow for further application such as a combination with numerical methods. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于简支厚板与薄板的关系

厚板的数学理论是建立在与薄板不同的力学假定的基础上的。本文分析了厚板与薄板之间静力学方面的关系。对于任意的简支多边形板,得到了厚板解通过薄板解的显式表达式,从而证明了:Reissner模型的厚板解与薄板解具有相同的剪力,但弯矩、转角、挠度有差别;而washizu模型的厚板解则与薄板解不仅剪力相同,连弯矩与转角亦相同,只是挠度有差别。     

Effect of layer geometry and stiffness on the fields of normal stresses in multilayer beams under asymmetric bending

A mathematical model is suggested for calculating the bending stiffness and fields of normal stresses (strength) at any point in the cross section of a multilayer beam. It is found that the structure of the scalar field of normal stresses allows one to solve some optimization problems with multivariant parameters. The method is illustrated with an example of two-layer beams. The results of an investigation into the strength and stiffness of two-layer beams, with a geometric and (or) stiffness asymmetry, in asymmetric bending are presented. The kinetics of bending stiffness and strength in relation to variations in the geometric parameters of cross sections and in the ratio of elastic moduli of layers is examined. It is established that the normal stresses in multilayer beams under asymmetric bending considerably depend on the location of the flexural center, neutral plane, and bending stiffnesses relative to the principal axes of cross sections of the beams.     

Asymptotic derivation of frictionless contact models for elastic rods on a foundation with normal compliance

Using asymptotic methods we derive some models for elastic rods in frictionless contact with a foundation with normal response. Starting from the three-dimensional problem we characterize the first terms of an asymptotic expansion of the solution taking the diameter of cross section as small parameter. Then we prove the convergence as this diameter tends to zero. In this way, we obtain and we mathematically justify a simplified model generalizing the best known classical models of such frictionless contact problems.     

Optimization of open cross sections of thin-walled beams with plate-shell-flanges

The paper is devoted to a monosymetrical cold-formed thin-walled beam with open cross section. Its flange consists of plates and circular shells. The beam is under pure bending. The cross section is characterized by dimensionless parameters. The authors are searching for an optimal cross section shape of considered beam. This optimal shape is determined by means of parametric optimization. The dimensionless objective function is so defined as to comprise both cross section area and a maximal allowable bending moment. The constraints follow from the local buckling conditions and geometric restrictions. Moreover, there are optimized cross sections of beams, for which the shear center is located either in the web or in the centroid of the cross section. Results are compared and analyzed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of linearly viscoelastic bars made from polymer materials of variable cross section

The critical viscoelastic forces for hinged bars made from polymer materials of variable cross section changing in accordance with a power law are obtained. It is shown that these forces are proportional to the critical elastic forces of bars of variable cross section.VMEL "Lenin," Sofia. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 540–546, May–June, 1973.     

中细柔性圆环壳整体弯曲的一般解及在波纹管计算中的应用(Ⅰ)——基本方程和一般解

讨论了圆环壳在横向载荷作用下的整体弯曲问题,提出了中细柔性圆环壳的概念、基本方程和一般解。本文称粗细比(子午线曲率半径与子午线曲率中心到回转轴的距离之比)小于1/3的圆环壳为中细圆环壳;方程的推导基于E.L.Axelrad的柔性壳理论;所得的解含有处理边界问题所必须的积分常数并且在全域内处处收敛;本解能方便地用于有关波纹管的计算。全文共分4个部分:(Ⅰ)基本方程和一般解;(Ⅱ)Ω型波纹管的计算;(Ⅲ)C型波纹管的计算;(Ⅳ)U型波纹管的计算。     

Dynamic modeling and analysis of rotating FG beams for capturing steady bending deformation

A dynamic analysis of rotating functionally gradient (FG) beams is presented for capturing the steady bending deformation by using a novel floating frame reference (FFR) formulation. Usually, the cross section of bending beams should rotate round the point at the neutral axis while centrifugal inertial forces are supposed to act on centroid axis. Due to material inhomogeneity of FG beams, centroid and neutral axes may be in different positions, which leads to the eccentricity of centrifugal forces. Thus, centrifugal forces can be divided into three componets: transverse component, axial component and force moment acting on the points of the neutral axis, in which transverse component and force moment can make the beam produce the steady bending deformation. However, this speculation has not been presented and discussed in existing literatures. To this end, a novel FFR formulation of rotating FG beams is especially developed considering centroid and neutral axes. The FFR and its nodal coordinates are used to determine the displacement field, in which kinetic and elastic energies can be accurately formulated according to centroid and neutral axes, respectively. By using the Lagrange's equations of the second kind, the nonlinear dynamic equations are derived for transient dynamics problems of rotating FG beams. Simplifying the nonlinear dynamic equations obtains the equilibrium equations about inertial and elastic forces. The equilibrium equations can be solved to capture the steady bending deformation. Based on the steady bending state, the nonlinear dynamic equations are linearized to obtain eigen-frequency equations. Transient responses obtained from the nonlinear dynamic equations and frequencies obtained from the eigen-frequency equations are compared with available results in existing literatures. Finally, effects of material gradient index and angular speed on the steady bending deformation and vibration characteristics are investigated in detail.