Elastic stability of orthotropic plates

A semi-numerical method is presented for the analysis of orthotropic rectangular plates subjected to uniform, linearly-varying and partial inplane loads. The solution procedure is based on the classical method of separation of variables. The basic functions are chosen as the eigenfunctions for straight prismatic beams in free vibration. The two-dimensional governing partial differential equation is first reduced to an ordinary differential equation. The classical unidirectional finite difference method is then employed to solve the resulting ordinary differential equation. Results are presented for plates with different edge and loading conditions.     

Buckling loads of variable thickness thin isotropic plates

This work presents accurate solutions for bifurcation buckling loads of rectangular thin plates with thickness that varies in the directions parallel to the two sides. The plates are subjected to biaxial compression and various combinations of boundary conditions are considered. The calculation of the critical loads was carried out by using the extended Kantorovich method. For the resulting ordinary differential equation, an exact method for the stability analysis of compressed members with variable flexural rigidity is used. The buckling load is found as the inplane load that makes the determinant of the stiffness matrix equal zero. New, exact results are given for many cases of uni-directional and bi-directional variation in thickness. The results are very accurate and exact for cases where an exact solution is available.     

Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM

A differential quadrature (DQ) procedure is developed for free vibration analysis of variable thickness moderately thick plates with edges elastically restrained against translation and rotation. The governing equations are based on the first order shear deformation theory and the rotary inertia effects are considered. Comparisons with known thin plate and uniform thickness Mindlin plate solutions are carried out to verify the applicability and accuracy of the analysis. Plates with linear or nonlinear varying thickness in one or two directions can be considered. It is demonstrated that using this DQ procedure, classical boundary conditions such as simply supported, clamped and free edges for variable thickness, thin as well as moderately thick plates can be simulated without any numerical difficulties. The effects of different combinations of constraints at edges, aspect ratio, thickness-to-length ratio, and stiffness parameter values on accuracy and convergence behaviors of the plates are presented. Some new results are presented for bi-linearly variable thickness plates with elastically restrained edges.     

Periodicity in the response of nonlinear plate, under moving mass

The dynamics of nonlinear thin plates under influence of relatively heavy moving masses is considered. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development of vibration amplitude, which is Duffing's oscillator with time varying coefficients. Through the application of Banach's fixed-point theorem, the periodic solutions are predicted. The method presented in this paper is general so that the response of plate to moving force systems can also be considered.     

三维层状路面结构动力响应的混合变量法

提出了三维多层路面结构动力响应分析的混合变量法和精细积分方法。通过Fourier-Bessel变换,将频率波数域内的偏微分波动方程解耦为两组二阶常微分方程,一组相应于P-SV波动分量,一组相应于SH波动分量。引入位移的对偶向量,将二阶常微分方程进一步简化为一阶齐次常微分方程,其解为指数函数,采用精细积分法可以获得很高的计算精度。此外,将解表示成混合变量形式,便于多层体系的合并和提高计算效率。该方法适用于任意荷载分布,计算稳定,便于计算机编程。将计算结果与BISAR软件结果及试验结果进行比较,验证了方法的精确性与合理性。在此基础上,分析了轮胎荷载作用下路面结构的变形特点及应力分布情况。研究结果可为公路路面结构设计提供参考。     

Bending analysis of thin annular sector plates using extended Kantorovich method

Highly accurate approximate closed-form solution is presented for bending of thin sector plates with clamped edges subjected to uniform and non-uniform loading using the extended Kantorovich method (EKM). Successive application of EKM converts the governing equation which is a forth-order partial differential equation (PDE) to two separate ordinary differential equations (ODE) in terms of r and θ. The obtained ODE systems are then solved iteratively with very fast convergence. In every iteration, exact closed-form solutions are obtained for both ODE systems. It is shown that the method provides sufficiently accurate results not only for deflections but also for stress components. Comparison of the deflection and stresses at various points of the plate shows very good agreement with results of other analytical and numerical analyses.     

任意厚度横观各向同性地基板的混合变分解

根据应力变分方程和伽辽金变分方程原理 ,推导出混合变分方程 ,并将其转换成状态方程 ,使状态空间理论和变分解相结合。将地基反力作为非独立状态变量直接引入到状态方程表达式中 ,给出横观各向同性地基厚板的变分解。对数值结果与精确解进行了对比 ,结果吻合较好     

正交各向异性桥面板比拟等效参数确定

借助弹性理论中经典的正交各向异性板的相应结论,依据响应等效原理,研究了构造正交各向异性桥面板等效为物理各向异性板中的参数确定问题,等效后的物理各向异性板在拉压刚度、抗弯刚度、动力特性以及质量等方面直接等效于原结构.算例表明,给出的等效参数确定过程有效可靠.该方法为建立大跨桥梁结构面向状态评估需求的高精度有限元模型奠定了基础.     

Non-linear vibration of initially stressed plates with initial imperfections

Non-linear partial differential equations of vibration for isotropic plates having initial imperfection are derived. The derivation based on the classical plate theory aims to describe non-linear vibration of imperfect plates in a general state of arbitrary initial stresses. Galerkin method is used to reduce the non-linear partial differential equations to ordinary non-linear differential equations. Runge-Kutta method is used to obtain the non-linear and linear frequencies of vibration. A numerical example is presented to discuss the performances of perfect and imperfect plates. The initial stress is taken to be a combination of pure bending stress plus an extension stress in the plane of the plate. It is found that the existence of initial vibration amplitude, initial stress and geometric imperfect may result in a drastic change on the non-linear vibration behavior. The effects of various parameters on the non-linear free vibrations are discussed.     

Buckling of centerline-stiffened plates subjected to uniaxial eccentric compression

A solution for the elastic buckling of flat rectangular plates with centerline boundary conditions subjected to non-uniform in-plane axial compression is presented. The loaded edges are simply supported, the non-loaded edges are free, and the centerline is simply supported with a variable rotational stiffness. The Galerkin method is used to establish an eigenvalue problem and a series solution for plate buckling coefficients is obtained by using combined trigonometric and polynomial functions that satisfy the boundary conditions. It is demonstrated that the formulation approaches the classical solution of a plate with a clamped edge as the variable rotational stiffness is increased. The variation of buckling coefficient with aspect ratio is presented for various stress gradient ratios. The coupling between plate aspect ratio, centerline rotational stiffness, and gradient of applied compressive stress is illustrated and discussed. The solution is applicable to stiffened plates and I-shaped beams that are subjected to biaxial bending or combined flexure and torsion, and is important to estimate the reduction in elastic buckling capacity due to stress gradient.     

Stability of variable thickness shear deformable plates—first order and high order analyses

This work presents analysis of the buckling loads for thick elastic rectangular plates with variable thickness and various combinations of boundary conditions. Both the first order and high order shear deformation plate theories have been applied to the plate's analysis. The effects of higher order non-linear strain terms (curvature terms) are considered as well. The governing equations and the boundary conditions are derived using the principle of minimum of potential energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the stability analysis of compressed members with variable cross-section, which provides for the derivation of the exact stiffness matrix of tapered strips including the effect of in-plane forces. The results from the two shear deformation theories are compared with those obtained by the classical thin plate's theory and with published results.     

Water flow in vertically stratified media

The transient and spatial water level in the soil surrounding a reservoir has been mathematically modeled and techniques of analog simulation presented. The model proposed assumes that the soil surrounding the reservoir is vertically stratified into regions, each with constant soil properties.

A change of variable which reduces the partial differential equation describing water flow in a region to an ordinary differential equation is presented. A trial and error method of analog simulation based on the reduced equations and specific boundary conditions is given and illustrated using a two-region example. Techniques of analog simulation for extensions of the two-region example are discussed.     

A new approach for free vibration analysis of nonuniform tall building structures with axial force effects

Dynamic analysis of beam‐like structures is significantly important in modeling actual cases such as tall buildings and several other related applications as well. This article studies free vibration analysis of tall buildings with nonuniform cross‐section structures. A novel and simple approach is presented to solve natural frequencies of free vibration of cantilevered tall structures with variable flexural rigidity and mass densities. These systems could be replaced by a cantilever Timoshenko beam with varying cross‐sections. The governing partial differential equation for vibration of a nonuniform Timoshenko beam under variable axial loads is transformed with varying coefficients to its weak form of integral equations. Natural frequencies can be determined by requiring the resulting integral equation, which has a nontrivial solution. The presented method in this study has fast convergence. Including high accuracy for the obtained numerical results as well. Numerical examples including framed tube as well as tube‐in‐tube structures are carried out in the study and compared with available results in the literature, and also with the results obtained from finite element analysis in order to show the accuracy of the proposed method in the study. Obtained results indicate that the presented method in this study is powerful enough for the free vibration analysis of tall buildings.     

Strength and ductility evaluation of steel bridge piers with linearly tapered plates

Important issues on the application of linearly tapered steel plates to highway bridge piers are discussed in this paper based on extensive experimental work. The factors that affect the seismic resisting performance of tapered plate steel columns were examined. The main factors considered are: (1) types of tapering mold line (outer, central, and inside tapering); (2) tapering ratio (or the slope of thickness change); (3) stiffeners’ rigidity; and (4) tapered stiffeners. To this end, four groups of specimens were tested under cyclic lateral loads and constant axial load. The columns with the outside taper plates were found to have the best strength and ductility characteristics among the three types of mold lines. In the case of different tapering ratios, the largest ductility was observed in the specimens with the tapering ratio slightly less than or equal to unity. The stiffeners’ rigidity ratio of around 3.0 was found to give the best ductility performance in columns with both constant thickness and varying thickness stiffeners. The test results were compared with a recently proposed design equation for limit strain of box columns with tapered plates. Finally, finite element analyses were performed and comparisons between the test and the analyses’ results of several test specimens are presented.     

Applications of diffusion models to reliability analysis of Daniels systems

Failure time is estimated for Daniels systems subject to quasistatic narrow band and dynamic broad band Gaussian load processes. Daniels systems are parallel systems with components (fibers) of independent indentically distributed resistances. The analysis is based on an approximate representation of response by diffusion models and probabilistic characteristics of the first passage time of these models. These characteristics include the mean and the probability of the first passage time and can be obtained respectively from a solution of an ordinary differential equation and by the path integral method. Results are given for simple Daniels system with two and three fibers.     

层状地基上预应力弹性板的竖向振动特性研究

基于解析层元法和经典弹性薄板理论,提出了一种求解层状地基与预应力弹性板竖向动力相互作用的计算方法。从地基的基本解和弹性薄板动力方程出发,借助Hankel积分变换,并结合地基与弹性板在接触面上的相容条件,得到变换域内地基与弹性板的动力耦合方程。通过求解耦合方程和数值逆变换,得到频域内预应力弹性板的位移解。与已有文献的结果进行对比,验证提出方法及计算程序的正确性及有效性。最后,通过算例分析,探讨了板–土刚度比和板径向预应力对预应力弹性板竖向动位移的影响。     

Analysis and shape optimisation of variable thickness prismatic folded plates and curved shells — Part 2: Shape optimisation

This paper deals with the development of computational tools for structural shape optimisation of shells and folded plates in which the strain energy or the weight of the structure is minimised subject to certain constraints. Both thickness and shape variables defining the cross-section of the structure are considered. The analysis is carried out using curved, variable thickness finite strips formulated and tested in Part 1 of this paper. Optimal shapes are presented for several shells and folded plates of variable thickness including plates on elastic foundation. The changes in the relative contributions of the bending, membrane and shear strain energies are monitored during the whole process of optimisation. The tools developed in the present work can be used as an aid to structural engineers in designing novel forms for shells and folded plates and provide valuable insight into the structural behaviour. It is concluded that the finite strip method offers an accurate and inexpensive tool for the optimisation of a wide class of structures having regular prismatic-type geometry with diaphragm ends.     

内衬波纹钢的钢筋混凝土壳体计算方法

根据经典壳体理论的柱壳微分平衡方程组，假设肋条、钢筋、波纹钢板的抗拉、抗弯作用均匀分布在肋条支持的范围内，忽略它们的抗扭作用，推导出了内衬三维波纹钢板的钢筋混凝土壳体的基本微分平衡方程组。采用三角级数法求解了均布荷栽作用下的壳体的中面位移，并与未衬三维波纹钢板的壳体进行了比较。通过比较可以看出，三维波纹钢板对于限制结构的变形有着十分明显的效果。     

Coupled thermoelasticity of functionally graded plates based on the third-order shear deformation theory

In this paper, the analytical solution is presented for a plate made of functionally graded materials based on the third-order shear deformation theory and subjected to lateral thermal shock. The material properties of the plate, except Poisson's ratio, are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The solution is obtained under the coupled thermoelasticity assumptions. The temperature profile across the plate thickness is approximated by a third-order polynomial in terms of the variable z with four unknown multiplier functions of (x,y,t) to be calculated. The equations of motion and the conventional coupled energy equation are simultaneously solved to obtain the displacement components and the temperature distribution in the plate. The governing partial differential equations are solved using the double Fourier series expansion. Using the Laplace transform, the unknown variables are obtained in the Laplace domain. Applying the analytical Laplace inverse method, the solution in the time domain is derived. Results are presented for different power law indices and the coupling coefficients for a plate with simply supported boundary conditions. The results are validated based on the known data for thermomechanical responses of a functionally graded plate reported in the literature.     

延拓Kantorovich法解薄扁壳弯曲问题

本文将Kerr提出的延拓Kantorovich法应用于薄扁壳的弯曲问题.文中取多项试探函数,由相应的变分原理导出了该问题的常微分方程组及其边界条件,利用常微分方程求解器COLSYS求解了大量的算例,其中包括薄板受集中力和多种曲率形式的薄扁壳受均布荷载的情况.数值结果表明,各种边界条件下的精度极高.