Why local damage increases the buckling load of circular plates

Local material damage is often characterised as degradation of material properties. A recent study has discovered a paradoxical phenomenon that clamped circular plates with a local axisymmetric damage may have a buckling load above that of a corresponding perfect plate. This paper describes a finite element investigation into the buckling behaviour of circular plates with a local axisymmetric damage. The paradoxical results of previous researchers are explained, and observations are made regarding the appropriate modelling of local damage.     

弹性半空间上圆板的弯曲分析

对各向同性弹性半空间地基上薄圆板一般弯曲问题进行了系统的分析.板的挠度、荷载、地基反力及板下地基表面的沉降均被展开为二重Fourier-Bessel级数,这些级数中的待定系数,由板的边界条件、板的控制方程及板-地基的相容条件加以确定,从而将弹性半空间地基与薄圆板的相互作用问题转化为数值积分和代数方程组的求解问题.数值计算表明,该级数解答具有较快的收敛速度.     

圆形结构下筏式基础变形反力分布规律试验研究

通过大型室内模型试验,指出圆形结构的荷载呈放射状传递到地基,当荷载与基础的相对挠曲呈线性关系时,柔性基础的基底反力、基础变形在结构外的2.5倍基础厚度范围内都是均匀分布的。柔性基础在破坏前的相对挠曲是逐渐增加的,而刚性基础在破坏时基础的相对挠曲突然增大。柔性基础的基底反力始终是中心点反力最大;刚性基础的中心点处基底反力小于周边基础反力。研究表明,基础变形、反力分布与基础和上部结构的刚度及形状有关,基础刚度越大,其调整基础变形、反力分布的能力越大。     

Effect of aspect ratio on the elastic buckling of uniaxially loaded plates with eccentric holes

The Finite Element Method (FEM) has been employed to determine the elastic buckling load of uniaxially loaded rectangular perforated plates with length a and width b. Plates with simply supported edges in the out-of-plane direction and subjected to uniaxial end compression in their longitudinal direction are considered. Integer plate aspect ratios, a/b=1, 2, 3 and 4, have been chosen to assess the effect of aspect ratio on the plate buckling load. Two perforation shapes of different sizes are considered; circular, and rectangular with curved corners. The rectangular perforation is oriented such that either its long or its short side is parallel to the longitudinal direction of the plate. The center of perforation was chosen at different locations of the plate. The study shows that the buckling load of a rectangular perforated plate that could be divided into equal square panels is not the same as that of the square panel that contains the perforation when treated as a separate square plate. For rectangular plates, the study recommends not to have the center of a circular hole placed in a critical zone defined by the end half of the outer square panel, to try always to put the hole in an interior panel of the plate, and to have the distance between the edge of a circular hole and the nearest unloaded edge of the plate not less than 0.1b. The study concludes also that the use of a rectangular hole, with curved corners, with its short dimension positioned along the longitudinal direction of the plate is a better option than using a circular hole, from the plate stability point of view.     

Solution for large amplitude vibrations of circular plates via modified Berger's approximation

Complex mathematical formulations for the geometrically nonlinear analysis of structural members have been presented by many researchers using the von-Karman strain-displacement relations. Berger proposed a simplifying approximation to take care of the geometric nonlinearity by neglecting the second strain invariant. Although it is a reasonable approximation, it poses problems when applied to circular plates. This is mainly due to the coupling between the radial and circumferential inplane strains, by the radial displacement. In this article, a modification to the Berger's approximation, applicable to the circular plates, is proposed. The main motivation of the present study is to examine how the modified Berger's approximation works in the case of the large amplitude vibrations of circular plates. It is shown that the modified Berger's approximation gives consistently accurate results, for both the simply supported and clamped circular plates, when compared to the classical results.     

A modified solution of radial subgrade modulus for a circular tunnel in elastic ground

In models based on Winkler springs for tunnel lining design, designers always face the difficulty of selecting appropriate values for the radial subgrade modulus (k_r). The widely used solution k_r for a circular tunnel in elastic ground proposed by Wood (1975) was found to be applicable only when the tunnel radial deformation is oval-shaped. On the basis of the Wood׳s solution, this note presents a general solution for k_r when the radial deformation of the tunnel is described by a Fourier series. This modified Wood׳s solution of k_r using compatible stress functions is validated by a numerical example. The modified solution for the example shows good consistency with the original Wood׳s solution when the tunnel becomes an oval shape with deformations. The example indicates that the magnitude of k_r is significantly affected by the distribution shape of the tunnel radial deformation. The value of k_r is no longer a constant value around the tunnel when the tunnel deforms into a general shape described by a Fourier series. It is quite different from the value of k_r for a distribution shape described by a single Fourier term, i.e. one involving a single frequency. The application of a general solution for k_r is illustrated by a design case using a bedded beam model.     

The effect of arching pressure on a vertical circular shaft

The lateral earth pressure of vertical circular shaft is investigated by using experimental tests and a theoretical analysis. The emphasis was quantifying on the magnitude and distribution of a lateral earth pressure measured by three-dimensional arching effect. A framework for determining the earth pressure distribution on the basis of both centrifuge model and full-scale field test results is introduced. Through these studies, it is found that the lateral earth pressure decreases within active displacement (δ_h/H: 0.2 × 10⁻², δ_h: horizontal displacement, H: Height of vertical shaft) of wall at deep excavation and the arching effect is more significant for deep excavation than for shallow excavation. It is also found that the lateral earth pressure acting on a vertical circular shaft considering arching effect is 80% smaller than that calculated by Rankine theory, and thus represents a significant improvement in the prediction of realistic earth pressure for vertical circular shaft subjected to arching pressure.     

Buckling of circular plates under intermediate and edge radial loads

This brief note is concerned with the elastic buckling problem of a circular plate subjected to both intermediate and edge radial loads. The stability criteria, in the form of transcendental equations, are derived as a function of the location of the intermediate load and the ratio of the magnitudes of the intermediate load and the edge load. Sample buckling results are presented for this new buckling problem.     

Finite element analysis of plates on elastic foundations: an automated approach

An automated approach for the analysis of plates on an elastic foundation is presented. The approach combines automatic mesh generation with nonlinear analysis to analyse plates of arbitrary shape on an elastic foundation subject to uplift. Two separate complete systems are presented, the first employing an ACM element to analyse rectangularly-shaped plates and the second employing a heterosis element to analyse plates of arbitrary shape. Solutions of the systems are presented and compared to theoretical results. A convergence study is also presented.     

Analysis of lateral earth pressure on a vertical circular shaft considering the 3D arching effect

The lateral earth pressure on a vertical circular shaft is investigated using both experiments and numerical analyses. The study focused on quantifying the magnitude and distribution of the lateral earth pressure, which was measured by considering the three-dimensional arching effect. A framework for determining distribution of the earth pressure based on centrifuge model tests and 2D FE analysis is introduced. The FE modelling techniques and the constitutive relationships of the soil are presented in detail. Parametric analyses showed that the arching effect on the lateral earth pressure is highly dependent on the diameter and height of the shaft, the internal friction angle and the cohesion value of the soil, the end-bearing conditions and the flexural modulus of the shaft. The study found that when the arching effect is considered, the lateral earth pressure on a vertical circular shaft is approximately 80% less than that calculated using Rankine’s theory. The study also found that the arching effect of the soil is more significant for flexible vertical shafts than for rigid vertical shafts embedded in weathered soil.     

高压旋喷成桩技术在软土路基沉降中的应用

分析了路基产生沉降的主要原因,并介绍了单管高压旋喷成桩技术在软土路基沉降中的应用,分别阐述了该技术的基本原理,设计参数的选择,施工工艺及加固效果,指出高压旋喷成桩技术控制沉降变形效果明显,值得推广。     

Elastic stability of bi-axially loaded rectangular plates with a single circular hole

The finite-element method has been employed to determine the elastic buckling stresses of a bi-axially loaded perforated rectangular plate with dimensions a and b in the x- and y-directions, respectively. The considered perforation is a single circular hole whose center is located along the longitudinal axis of the plate. The considered plate has simply supported edges in the out-of-plane direction and is subjected to bi-axial uniformly distributed end loads (compressive load σ_x in the x-direction, and compressive or tensile load σ_y in the y-direction). Parameters considered in the study are the plate's aspect ratio a/b, the stress ratio ξ between the applied stresses in the y- and x-directions (ξ=σ_y/σ_x), the circular hole size d and location e_x.The study shows that, in most of the considered cases, the bigger the hole size d, the lesser the plate stability and the lesser the buckling stresses. It also shows that the plate aspect ratios, a/b, between 0.6 and 1.2 should be avoided for plates with large holes and negative ξ, due to the large reduction in the buckling stresses. The hole location should also be selected to be away from the loaded edge of the plate as much as possible (better to have e_x/b>1.0) to increase the buckling stresses and improve stability. The study demonstrates also that the increase in tension in the y-direction in bi-axially loaded plate with large hole (d/b>0.4) reduces its stability. This is in contrary to the expected increase in the stability due to the increase in tension which can be seen clearly in the cases of solid plates and plates with small holes.     

On anisotropic continuous damage of plates in Flexure

Internal damage due to fissuration results in an overall anisotropic material behaviour. A scalar damage parameter does not allow one to model a direction dependent response of continuously damaging solids. Employing the idea that the scalar damage parameter can be associated with an appropriate strain a damage tensor related to strains is introduced. The overall elastic properties are determined using a simplified form of the sensor function representations and the stress-strain relations regarding the overall response are given. For combined stress states an uncoupled damage theory is derived for materials with no lateral deformability under axial stress. Both brittle-ductile and elastic-brittle cases are studied using circular plates, for the elastic-brittle case only the governing differential equation are presented.     

Buckling of circular plates with an internal ring support and elastically restrained edges

Presented herein are the exact buckling solutions of circular plates with a concentric internal ring support and elastically restrained edges. This study shows the existence of buckling mode switching with respect to the radius of the internal ring support. Generally the plate buckles in an axisymmetric mode but when the ring support radius becomes small, the plate may buckle in an asymmetric mode. The cross-over ring support radius varies from 0.081 to 0.152 times the plate radius, depending on the rotational stiffness of the elastic restraint at the edges.     

高耸构筑物厚地基圆板的混合变分解

根据应力变分和位移变分原理,导出圆板的混合变分方程,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人到状态方程表达式中,得出任意厚地基圆板的变分解,同时给出了数值结果。     

钢筋混凝土圆管涵管周土压力的试验研究

对具有刚性座垫的钢筋混凝土圆管涵进行了现场试验研究，测试出不同填土高度下的管周土压力、刚性座垫的支座反力和车辆活载引起的管周土压力，并与国内外主要规范进行了比较分析。     

Vibrations of rectangular plates with elastic intermediate line-supports and edge constraints

A set of static beam functions, which are the solutions of an elastically point-supported beam under a Fourier series of static sinusoidal loads distributed along the length of the beam, are developed as the admissible functions to analyze the vibrations of orthotropic rectangular plates with elastic intermediate line-supports using the Rayleigh–Ritz method. Both the elastic rotational and the elastic translational constraints along the edges of the plate are also considered simultaneously. Unlike conventional admissible functions, this set of static beam functions not only can automatically adjust to the stiffnesses of the intermediate line-supports but also can properly describe the discontinuity of shear forces at the line-supports so that higher accuracy and faster convergence can be expected for the dynamic analysis of such plates. The suggested approach is effective even for various limiting cases by letting the corresponding stiffnesses approach their natural limits of zero or infinity. The present method is theoretically sound and mathematically simple, with each of the static beam functions being only a third-order polynomial plus a sine function. A common and efficient computational program can be compiled because of the fact that a change of the line-support parameters (locations, number and stiffnesses) and the boundary conditions of the plate only results in a corresponding change of the coefficients of the polynomial in the static beam functions. Several numerical examples are presented and the results obtained, where possible, are compared with the known solutions in literature. The present method has proved to be extremely effective for solving the aforementioned problems.     

轴对称受荷弹性圆板下横观各向同性多层地基分析

将地基视为多层横观各向同性体,基于弹性圆板与地基的竖向位移协调条件与光滑接触条件,结合多层横观各向同性地基应力与位移的传递矩阵解,推导出多层横观各向同性地基上轴对称受荷弹性圆板问题的解析解。编制了相应的计算程序,并对数值计算结果进行了对比分析。结果表明：圆板刚度的增大有助于减小板底位移和沿深度竖向应力;地基横观各向同性性质对板底位移及板中心下沿深度竖向应力影响显著。