Stability of bars with variable rigidity

A variable cross-section bar is considered. The bar is not uniform in length. The bar axis through the mass centers of all cross sections is a straight line. The bar is compressed by a longitudinal force applied to the mass center of the boundary cross section. The stability loss of the straight-line shape of the bar’s equilibrium is discussed when a curved shape is also possible. Approximate analytical formulas are obtained for the critical compressive force when four types of end fixing are used for a periodically nonuniform bar. The numerical results obtained by these formulas are compared with the known exact solutions to the stability equation for a bar whose cross section is stepwise variable and whose nonuniformity consists of only one period (the limiting case).     

Natural frequencies of longitudinal oscillations for a nonuniform variable cross-section rod

The natural frequencies of longitudinal oscillations of a rod such that its Young’s modulus, the density, and the cross-sectional area are functions of the longitudinal coordinate are analyzed. For solving the corresponding problem, an integral formula is used to represent the general solution to the original Helmholtz equation with variable coefficients in terms of the general solution to the accompanying equation with constant coefficients. Frequency equations are derived in the form of rapidly converging Leibniz series for three types of boundary conditions. For these cases the frequency zerothapproximation equations are given to quickly find the lowest natural frequencies with an adequate accuracy.     

NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE——REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅱ)

Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible proliferation.The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation explicity generation by generation.As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincare problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds of singularities in a unifying way.Exact explicit analytic expression for irregular integrals can be obtained by means of correspondence principle.It is not difficult to prove the convergence of the tree series solution obtained. Direct substitution shows it satisfies the equation.The tree series is automorphic, which agrees completely with Poincaré’s conjecture.     

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].One of the methods for solving the equations with rapidly oscillating coefficients is the asymptotic method of small geometric parameter. The standard procedure of this method usually includes preparatory stages. At the first stage, as a rule, a rectangular periodicity cell is distinguished to be a typical element. At the second stage, the scale of global coordinates is changed so that the rectangular structure periodicity cells became square cells of size l × l. The third stage consists in passing to the dimensionless global coordinates relative to the plate characteristic dimension L. As a result, the dependence between the new local variables and the new scaled dimensionless variables is such that the factor 1/α, where α=l/L ? 1 is a small geometric parameter, appears in differentiating any function of the local coordinate with respect to the global coordinate. After this, the solution of the problem in new coordinates is sought as an asymptotic expansion in the small geometric parameter [1], [4–10].We note that, in the small geometric parameter method, the asymptotic series simultaneously have the form of expansions in the gradients of functions depending only on the global coordinates. This averaging procedure can be applied to linear and nonlinear boundary value problems for differential equations with variable periodic coefficients for which the periodicity cell can be affinely transformed into the periodicity cube. In the case of an arbitrary dependence of the coefficients on the coordinates (including periodic dependence), another averaging technique can be used in linear problems. This technique is based on the possibility of the integral representation of the solution of the original problem for the linear equation with variables coefficients in terms of the solution of the same problem for an equation of the same type but with constant coefficients [11–13]. The integral representation implies that the solution of the original problem can be represented in the form of the series in the gradients of the solution of the problem for the equation with constant coefficients [13].The aim of the present paper is to develop methods for calculating effective characteristics of corrugated plates. To this end, we first write out the equilibrium equations for a flexible anisotropic plate, which is inhomogeneous in the thickness direction and in the horizontal projection, with an initial deflection. We write these equations in matrix form, which allows one to significantly reduce the length of the expressions and to simplify further calculations. After this, we average the initial matrix equations with variable coefficients. The averaging procedure implies the statement of problems such that, after solving them, we can calculate the desired effective characteristics. By way of example, we consider the case of a corrugated plate made of a homogeneous isotropic material whose corrugations are hexagonal in the horizontal projection. In this case, we obtain approximate expressions for the components of the effective tensors of flexural rigidity and longitudinal compliance and expressions for the effective plate thickness.     

Solution of flat crack problem by using variational principle and differential–integral equation

Variational principle is used to solve some flat crack problems in three-dimensional elasticity. In the formulation, the strain energy is evaluated by multiplying the crack opening displacement (COD) by the boundary traction. The boundary traction is related to the COD function by a differential–integral representation. By using an integration by part, the portion of the strain energy of the potential functional can be expressed by a repeated integral. In the integral all the integrated functions are non-singular. Letting the functional be minimum, the solution is obtained. In the actual solution, the COD function is represented by a shape function family in which several undetermined coefficients are involved. Using the variational principle, the coefficients are obtained. Several numerical examples are given with the stress intensity factors calculated along the crack border.     

The complex variable boundary element method for solving sandwich plate problems

The objective of this paper is to develop a new complex variable boundary element method for sandwich plates of Reissner's type and Hoff's type. The general solution of Helmhotz equation in complex field is given. Based on the Vekua's complex integral representation of the analytic function, the new boundary integral equations are formulated. The density function in the integral equation is determined directly by boundary element method. Some standard examples are presented, and the results of numerical solutions are accurate everywhere in the plate. The approach presented is only applicable for bounded simply connected regions. The project is supported by the National Science Foundation of China.     

夹层板的复变边界元解法

本文用全纯函数表示微分方程△f(x,y)-λ(~2)f(x,y)=0的一般解,粮据全纯函数的Bekya积分表示法,建立了复数域内的边界积分方程并针对各种边界条件下Reissner型夹层板、Hoff型夹层板进行了数值求解。     

分布轴向力作用下隔水管横向自由振动分析

隔水管固有频率的精确计算对保证隔水管的安全使用和防止共振的发生有着极为重要的意义.在分析中,考虑了分布轴向力和顶张力的共同作用,建立了隔水管横向振动力学模型;基于牛顿定律和纵横弯曲梁理论,对微单元受力分析,得到隔水管横向自由振动的四阶偏微分方程;利用分离变量法将四阶偏微分方程简化为四阶变系数常微分方程;采用积分法求解四阶变系数常微分方程,得到隔水管横向自由振动固有频率的解析解.结果表明:(1)分布轴向力作用下隔水管横向自由振动的固有频率和振型,与将分布轴向力简化为集中力作用下隔水管的固有频率和振型有很大差别;(2)顶张力一定时,随着分布轴向力减小,隔水管固有频率增大;分布轴向力一定时,随着顶张力增大,隔水管固有频率增大;(3)采用积分法求解隔水管横向振动特性时,计算精度高,为隔水管的优化设计提供了可靠的理论依据.     

Buckling load optimization of beams

In this paper, shape optimization is used to optimize the buckling load of a Euler–Bernoulli beam having constant volume. This is achieved by varying appropriately the beam cross section so that the beam buckles with the maximum or a prescribed buckling load. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds. The evaluation of the objective function requires the solution of the buckling problem of a beam with variable stiffness subjected to an axial force. This problem is solved using the analog equation method for the fourth-order ordinary differential equation with variable coefficients. Besides its accuracy, this method overcomes the shortcomings of a possible FEM solution, which would require resizing of the elements and recomputation of their stiffness properties during the optimization process. Several example problems are presented that illustrate the method and demonstrate its efficiency.     

A necessary and sufficient set of boundary integral equations with indirect unknowns for plate bending problem

A boundary integral representation of plane biharmonic function is established rigorously by the method of unanalytical continuation in the present paper. In this representation there are two boundary functions and four constants which bear a one to one correspondence to biharmonic functions. Therefore the set of boundary integral equations with indirect unknowns based on this representation is equivalent to the original differential equation formulation.     

The concept of material forces in phase transition problems within the level-set framework

The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.     

Quasi-Green＇s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.     

Stability of the in-plane bending mode of thin-walled framed structures

The paper outlines an approach to solving the stability problem for framed structures under arbitrary transverse loading. The available methods are limited by one law of variation in the bending moment responsible for loss of stability. The equilibrium equations for a thin-walled bar are integrated assuming that the bending moment is constant. The solution of the Cauchy problem is given in normal form. The arbitrary varying bending moment is approximated by a piecewise-constant function, which will be a little different from the original if the bar is partitioned into a great number of segments. The equations of the boundary-value problem for a discretized framed structure are derived using the boundary-element method. The critical forces and moments are determined from a transcendent equation. Numerical solutions are presented to demonstrate the high accuracy and efficiency of the approach. The solutions of test problems are in agreement with those obtained by Timoshenko     

Longitudinal free vibration of inhomogeneous elastic straight strut with a variable cross section

The scope of the present article, motivated by the case of the composite wooden propeller of an airplane, is to deal tentatively with the longitudinal free vibrationproblem of an elastic straight bar with a more general mathematical treatment.In this analysis, we have assigned to the modulus of elasticity, the bar cross section as well as the mass per unit length of the bar an exponential function variation, and then found a general solution, wherein three parameters were considered as the main factors to affect the longitudinal free vibration of the inhomogeneous elastic straight bar with a variable cross section.     

Dynamic penny-shaped cracks in multilayer sandwich composites

A point force method is proposed for obtaining the dynamic elastic response of a multilayer sandwich composite in the presence of a penny-shaped crack under a harmonic loading. The sandwich composite is a multilayered solid whose lower half is the mirror image of the upper half with the center plane as the mirror. The crack is lying on the mirror plane of the composite. The solution of the mode I dynamic crack problem is formulated by integrating the Green’s function of a time-harmonic surface normal point force over the crack surface with an unknown point force distribution. The dual integral equations of the unknown point force distribution are established by considering the boundary conditions, which can be reduced to a Fredholm integral equation of the second kind. A complete solution of the crack problem under consideration can be obtained by solving this Fredholm integral equation. It will be shown that the results obtained by this approach are the same as some existing solutions.     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

A GENERAL FORMULA FOR CALCULATING FORCES ON A 2-D ARBITRARY BODY IN INCOMPRESSIBLE FLOW

In the present paper, a general integral equation is presented to calculate the forces exerted on a two-dimensional (2-D) body of arbitrary shape immersed in unsteady, incompressible flows. By finding the general solutions of a set of Laplace equations with particular boundary conditions, the equation can be simplified to produce a simplified formula for calculating the forces. The simplified formula consists of three parts, representing contributions from different physical phenomena: added mass force and/or inertial force in inviscid flow, the force caused by the deformation of fluid and viscosity and the force caused by the convection of fluid with nonzero circulation. It can be applied to any 2-D arbitrary body in viscous or inviscid, steady or unsteady incompressible flow. As the formula excludes either temporal derivatives of velocity or spatial derivatives of vorticity in the flow field, the numerical errors contained in the numerical solution of velocity and vorticity fields will not be magnified, and therefore the resulting force calculated is more accurate. Most importantly, the formula presents an alternative method for obtaining the added mass of a 2-D body of arbitrary shape accelerating in a fluid. For bodies of simple shape, such as a circle, ellipse and plate, the added masses predicted using the present method are in agreement with that obtained by conventional methods. For bodies of complex shape, the present method only requires the calculation of the first two coefficients of the conformal transformation and cross-sectional area.     

非圆截面杆中非线性扭转波方程的解析求解

研究了非圆截面杆中非线性扭转波动方程的精确求解问题. 利用直接积分与微分变换相结合的方法,得到了该方程的隐式通解. 通过对积分常数和方程系数的不同情形的讨论, 给出了该方程的三角函数、双曲函数、椭圆函数、指数函数以及它们的组合形式的解,分别对应于的非线性扭转波的孤立波、周期波以及冲击波等多种传播形式.     

A NEW ENGINEERING CALCULATING MODEL FOR A LINEAR INCLUSION AND ITS APPLICATIONS

By using the hypothesis of the deformation of the straight bar and beamin mechanics of materials,a new engineering calculating model for a linear inclusion inplane is presented.Through the Kelvin's solution of a concentrated force,the inclusionproblem is reduced to solving a set of uncoupled singular integral equations which canbe solved by the numerical method of singular integral equation.Based on theseresults,several applicable examples including an inclusion-crack problem are calculatedand the results are quite satisfactory.     

基于S变换的非平稳随机过程演变功率谱密度估计

提出了一种基于S变换的估计Priestley非平稳随机过程演变功率谱密度的方法。此方法的根本在于,相对于S变换的"变换核",Priestley非平稳随机过程的调制函数为慢变函数。因此,非平稳随机过程的S变换可视为相位修正后的另一非平稳随机过程。推导出了对应于特定频率点的S变换瞬时均方值和非平稳随机过程演变功率谱密度之间的关系式。将功率谱密度函数表达为有限个频率点的级数展开,通过求解一组代数方程,就能得到级数展开中每个频率点的时变系数,由此,可给出非平稳随机过程的演变功率谱密度。由于级数展开中的高斯形状函数不依赖于时间,因此,本文所提算法具有较高的计算效率。最后,给出了均匀调制和非均匀调制非平稳随机过程演变功率谱估计的算例。