一种分析微悬臂梁大挠度变形的新算法

考虑静电力边缘效应的影响,建立了微悬臂梁大挠度变形的静态变形分析模型,通过梁弯曲理论将控制方程化为一阶非线性微分方程组,结合非线性方程组求根、迭代修正齐次扩容精细积分法和增量迭代法,提出了一种分析微悬臂梁大挠度变形的半解析、半数值算法。数值算例表明:该方法具有较高的精度和稳定性,是分析微悬臂梁变形的一种有效方法;通过对微梁静态特性的分析表明:考虑边缘效应后微梁的吸合电压减小,考虑大挠度变形的影响后微梁的吸合电压增加。     

An incremental analysis of beam-columns and frames including finite deformations and bilinear elasticity

A consistent incremental method is presented for the finite deformation analysis of beam-columns and simple plane frames in the elaslo-plastic range. The method has made it possible to examine the effects of previously neglected terms in the tangent flexibility matrix for a cross section. These terms, the off-diagonal ones, relate increments of bending moment and axial load to changes in centroidal strain and curvature respectively. Formulae and values are given for the flexibility coefficients for a rectangular cross section of a bilinear strain hardening material. The problems of local unloading within the section, i.e. decrease in stress in part of the section when bending moment or axial load increase, and local unloading in frames are discussed in some detail.The results of a series of computer analyses for beam-columns which explored the effects of the off-diagonal flexibilities are given, and an experimental programme in which several rigid jointed two-bar frames made from three different steels were tested and compared with the present theory is reported.     

Microcantilever-based weather station for temperature,humidity and flow rate measurement

The current study develops a new process for the fabrication of Pt resistor temperature detectors (RTDs), cantilevers covered with a water-absorbent polyimide layer for humidity measurement and the bending-up of cantilevers to determine the flow rate. Pt RTDs are fabricated on the silicon substrate. The temperature measurement is based on the linear resistance variations when temperature changes. The polyimide layer is spun on the cantilever to form a humidity sensing layer. A variation in humidity causes moisture-dependent bending of the micro-cantilever, which changes the measured resistance of the resistor on the micro-cantilever. The same type of micro-cantilever, without spinning on polyimide, is used to form an anemometer. It is found that the cantilever bends slightly upward as a result of the released residual stress induced in the beam during the fabrication. When wind passes over the cantilever beam, a small deformation occurs. Variations in the flow rate can therefore be determined by measuring the changes in resistance caused by the beam deflection, using a LCR meter.     

The large amplitude vibration of hinged beams

The large amplitude free vibrations of a simply-supported beam with ends kept a constant distance apart is studied using the actual nonlinear equilibrium equations (i.e. specification of loads in terms of the deformed coordinates of the beam) and the exact nonlinear expression for curvature in addition to the nonlinearity arising from the axial force. A variable separable assumption, together with certain assumptions as to the behaviour of the time function defines an eigenvalue characteristic of the vibration. A numerically exact successive integration and iterative technique establishes the dependence of this quantity on the amplitude of vibrations. The hardening effect of nonlinearity is then interpreted in terms of the variation of this quantity with the amplitude of vibration. This new criteria to define nonlinearity, is compared with several existing in the literature. The present analysis allows the separation of the effects of stretching and large deflection equations on the nonlinear behaviour and the conclusion can be made, based on numerical evidence, that the predominant nonlinearity is due to stretching. The axial force at any station in the beam and the bending stress can also be computed in a numerically exact sense, at the point of maximum amplitude.     

Dynamic response analysis of stiffened slab bridges

The dynamic response of skew bridge decks with stiffeners has been investigated using higher order finite strip method. The eccentricity of the stiffeners is also considered in the analysis. Normal mode method in conjunction with William's method to accelerate the convergence of the solution has been used to find the dynamic response of the bridge deck due to moving load. The influence of eccentricity of stiffener on the deflection and bending moments in the deck, and bending moment in the beams, has been investigated. Numerical work has been presented for different skew angles and the speeds of the moving force.     

Nonlinear analysis of steel frameworks through direct modification of member stiffness properties

This paper presents a computer-based method for nonlinear analysis of planar steel frameworks under monotonic loading that is directly based on the matrix displacement method of analysis. Stiffness degradation factors progressively deteriorate the post-elastic bending, shearing and axial stiffness properties of framework members over an incremental load history until failure of part or all of the structure occurs. The analytical procedure is based on Timoshenko beam theory to allow for the analysis of frameworks for which effects of shear deformation on elastic behavior may be significant, and employs transcendental stability functions to model the effect of axial force on the bending stiffness of members. Also accounted for is the influence of residual stresses on the initial-yield and full-yield capacities of members, and the effect of both combined moment plus axial force and combined moment plus shear force on the post-elastic behavior of members. The nonlinear analysis method is applied for two benchmark planar steel structures, and it is shown to give results comparable to both experimental and analytical results previously published in the literature. The method is readily implemented on a computer using conventional matrix structural analysis techniques, and is directly applicable for the efficient nonlinear analysis of frameworks of any scale or complexity.     

悬索桥主缆初应力对桥梁动力特性的影响

为掌握主缆初应力对桥梁动力特性的影响,以悬索桥为例,采用静力非线性分析方法计算应力变化过程中悬索桥的跨中挠度、缆索轴力及加劲梁的轴力变化;将得到的内力作为结构的预加应力进行有预应力的模态分析. 应用ANSYS软件进行分析,其中有限元建模时采用4种单元类型:空间梁单元BEAM4用于模拟加劲梁和塔;空间杆单元LINK10用于模拟主缆及吊杆;通过设定BEAM4和LINK10单元初应变进行有预应力的模态分析;采用MASS21单元模拟横隔板、吊杆锚固装置和桥面上的栏杆,并分别考虑质量和质量惯性矩的作用. 分析表明,悬索桥主梁竖弯振动频率受主缆初应力的影响较大,而侧弯振动频率和扭转振动频率受此影响较小. 该结果为同类桥梁的动力特性分析提供参考.     

Nonlinear static and dynamic damped response of isotropic, elastic circular plates

Nonlinear static and dynamic response analysis of clamped-in and simply supported boundary conditions, and immovably constrained and stress-free edge conditions for circular plates of isotropic elastic material with damping, subjected to step pressure pulse excitation are presented. The Von Karman relations are used which are reduced to coupled nonlinear partial differential equations and solved by a one-term solution, applying the Ritz-Galerkin technique to the deflection equation. This yields an ordinary nonlinear differential equation in time. The nonlinear dynamic damped response is obtained by applying the ultraspherical polynomial approximation technique. Plots of static deflection to thickness ratio versus non-dimensional load for different boundary and edge conditions, and the effect of damping on the nonlinear dynamic response for different values of non-dimensional damping factor are presented.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.     

Deflection of elastic-plastic frames at finite spread of yielding zones

In elastic-perfectly plastic beams and frames finite zones of yielding develop under the collapse load. The classical deflection analysis assumes that the elastic rigidity reduces locally to zero at the yield hinges only. Such an approach underestimates the deflections at collapse.A method is proposed in order to evaluate the displacements of elastic-plastic frames when the actual spread of plastic zones is included in the analysis. In comparison with the classical method the developed technique accounts for the flexural stiffness variation between the hinges and due to the partial yielding of cross-sections. The actual extent of plastic zones depends on the bending moment distribution in the structure.The method contains both the algorithms and program for the deflection evaluation, specified for I-shape cross-sections. The elastic-plastic moment-curvature relations are derived and integrated to obtain additional rotations due to the flexural stiffness variation when the structure transforms in a mechanism. A linear programming program developed earlier is modified to account for the supplementary rotations. The essential features of the method and those of the program application are illustrated purposely in simple examples, showing the order of magnitude of the actual displacements in comparison with the results of the classical deflection analysis of elastic-plastic beams and frames.     

Arch shape optimization using force approximation methods

The objective of this paper is to provide a method of optimizing areas of the members as well as the shape of both two-hinged and fixed arches. The design process includes satisfaction of combined stress constraints under the assumption that the arch ribs can be approximated by a finite number of straight members.In order to reduce the number of detailed finite element analyses, the Force Approximization Method is used. A finite element analysis of the initial structure is performed and the gradients of the member end forces (axial, bending moment) are calculated with respect to the areas and nodal coordinates. The gradients are used to form an approximate structural analysis based on first order Taylor series expansions of the member end forces. Using move limits, a numerical optimizer minimizes the volume of the arch with information from the approximate structural analysis.Numerical examples are presented to demonstrate the efficiency and reliablity of the proposed method for shape optimization. It is shown that the number of finite element analysis is minimal and the procedure provides a highly efficient method of arch shape optimization.     

Thermal post-buckling characteristics of clamped skew plates

Thermal post-buckling characteristics of clamped skew plates with restrained edges subjected to planar temperature distributions are studied. The problem is formulated in terms of Von Kármán equations expressed in terms of displacements generalised to include thermal effects. A perturbation method is employed to obtain a linear set of partial differential equations. The solution to this linear set is then obtained by using the Galerkin method. Numerical results are presented for different skew plate configurations for both uniform and two-dimensional temperature distributions. Plots of central deflection, membrane and bending stress distributions are presented and the post-buckling characteristics are discussed in detail.     

The finite element solution of a class of elastica problems

A finite element procedure for the analysis of an inextensional elastica bent through frictionless supports is presented. Element displacements are expressed in terms of cubic Hermite polynomials with nodal displacements and derivatives being determined to minimise the strain energy. It is shown that the axial force at any point is proportional to the square of the bending moment, which enables member incremental stiffness matrices to be expressed in a similar form to those used in stability analysis. The iterative procedure proposed for solving the nonlinear stiffness equations may readily be incorporated into any continuous beam program by the modification and addition of very few statements. An example is considered which indicates that accurate solutions may be obtained to a wide range of practical problems using the proposed technique.     

Symbolic and Numeric Methods for Exploiting Structure in Constructing Resultant Matrices

Resultants characterize the existence of roots of systems of multivariate nonlinear polynomial equations, while their matrices reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse (or toric) resultant, which takes into account the sparse structure of the polynomials. The construction of sparse resultant, or Newton, matrices is the critical step in the computation of the multivariate resultant and the solution of a nonlinear system. We reveal and exploit the quasi-Toeplitz structure of the Newton matrix, thus decreasing the time complexity of constructing such matrices by roughly one order of magnitude to achieve quasi-quadratic complexity in the matrix dimension. The space complexity is also decreased analogously. These results imply similar improvements in the complexity of computing the resultant polynomial itself and of solving zero-dimensional systems. Our approach relies on fast vector-by-matrix multiplication and uses the following two methods as building blocks. First, a fast and numerically stable method for determining the rank of rectangular matrices, which works exclusively over floating point arithmetic. Second, exact polynomial arithmetic algorithms that improve upon the complexity of polynomial multiplication under our model of sparseness, offering bounds linear in the number of variables and the number of non-zero terms.     

Numerical analysis of the dynamic effects of shock-load-induced ice shedding on overhead ground wires

A numerical model using nonlinear finite element analysis is proposed to calculate the dynamic effects of ice shedding induced by a pulse-type excitation on a single-span overhead line section. The excitation simulates the effect of an external load intended to remove the accreted ice from the cable. Several ice-shedding scenarios are studied with variables including span length and pulse-load characteristics. Different pulse-load characteristics are represented by the variation of their time histories while their amplitude is kept constant. This model serves as a basis to study various failure criteria of atmospheric glaze ice in terms of stress–strain relations and strain-rate effects. The failure criteria defined for glaze ice incorporate both axial and bending effects.     

Large elastic deformations of clamped skewed plates by dynamic relaxation

This investigation is concerned with the nonlinear behaviour of clamped Isotropic skew plates of constant thickness subjected to a uniformly distributed transverse load. The recently developed numerical technique of dynamic relaxation has been adopted for the analysis. A detailed study of the large deflection behaviour of skew plates has been made by varying three parameters, viz. skew angle, load, and aspect ratio. Numerical results have been compared with the available solutions. Representative nondimensional solutions are presented in the form of graphs to elucidate the nonlinear effect due to large deflection at higher loads.     

Large deflection analysis of clamped skew sandwich plates by parametric differentiation

The differential equations governing the large deflection behavior of skew sandwich plates are highly complex in nature and do not lend themselves for easy solution. In the study reported herein, this problem is solved by a numerical technique known as “parametric differentiation”. The non-linear differential equations in terms of displacements are transformed into a set of linear differential equations with variable coefficients. These are solved to give the gradients of the displacements in the load direction. The subsequent solution of a set of initial value problems yield the displacements proper. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load deflection characteristics have been presented for clamped skew sandwich plates.     

Ultimate strength analysis of prestressed reinforced concrete sections under axial force and biaxial bending

A secant approach is illustrated for the ultimate limit state (ULS) analysis of prestressed reinforced concrete sections subjected to axial load and biaxial bending in presence of softening stress–strain laws. The stiffness matrix and the resultant loads are evaluated analytically by a novel methodology, termed fiber-free, which represents a computationally efficient alternative to fiber approaches. Extensive computations of the ULS domains of benchmark test cases show that the robustness of the proposed algorithmic strategy is substantially unaffected by the amount of reinforcement, prestressing and softening, though localized non-convex regions have been occasionally experienced in presence of softening.     

Yield surface linearization in elastoplastic analysis

A method for the historical elastoplastic analysis of plane frames under the assumption of small deformation theory is presented. Interaction of bending moment and axial force is taken into account. A major feature of the procedure lies in the piecewise linearization of the convex nonlinear yield surface, thereby guaranteeing the satisfaction of both equilibrium and yield conditions. Local unloading can also be catered for. A computer program based on this technique has been written and three simple numerical examples are given to illustrate its capabilities.     

Nonlinear finite element analysis of elastic frames

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.