Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

双轴载荷作用下源于椭圆孔的分支裂纹的一种边界元分析

利用一种边界元方法来研究双轴载荷作用下无限大板中源于椭圆孔的分支裂纹．该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者提出的裂尖位移不连续单元构成．在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界,文中算例说明本数值方法对计算平面弹性裂纹的应力强度因子是非常有效的。该文对双轴载荷作用下无限大板中源于椭圆孔的分支裂纹的数值结果进一步证实本数值方法对计算复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示双轴载荷及裂纹体几何对应力强度因子的影响。     

Photoelastic analysis of an asymmetrically reinforced circular cut-out in a flat plate subjected to uniform unidirectional stress

Three-dimensional photoelastic studies of stresses around an asymmetrically reinforced circular cut-out in a flat plate under uniform unidirectional stress are reported. The frozen-stress technique, with Hysol 4290 material, was used to determine the stress distribution through four critical points on the boundary of the reinforced hole. Included were models with different cross sections of reinforcement, with various interface fillet radii and with different plate widths. For the majority of models, the ratio of volume of reinforcement to volume of hole was unity. It is concluded that, for reducing the stress concentration, there is a limit on the effectiveness of increasing the fillet radius beyond half the plate thickness. It was found that a reinforcement having a thickness of approximately 40 percent of the plate thickness was optimum and that the stress concentration decreases with volume of reinforcement. The authors believe that, with judgment, some of the conclusions reached may be applied, for design purposes, beyond the specific dimensional ranges and loading conditions of the tests.     

A multilayer-reflection technique for three-dimensional photoelastic studies of perforated plates in bending

A photoelastic investigation was conducted to determine the stress-concentration factors around a large, symmetrically reinforced central hole in a square plate under 1∶1 and 2∶1 biaxial bending. Tapered-edge rings served as the reinforcement, and a major objective was to determine the ring proportions such that the maximum stress at the hole would be equal to the value which would be present in an unperforated plate under the same nominal stress. Because the stress distribution at the periphery of a hole in such a plate structure varies in the radial, tangential and thickness direction, it was necessary to employ a three-dimensional photoelastic technique. There were a number of serious disadvantages in the use of any of the standard procedures and a new three-dimensional technique for room-temperature use was developed which is particularly suitable for the determination of boundary stresses around holes in bending experiments. With the technique in its present state of development, the three-dimensional isochromatic distribution in the plate can be determined from a single model and, from this, the boundary value of stress. The new technique utilized a laminated-plate model. Selective aluminizing of the laminations allowed for the determination of fringe-order distributions in the thickness direction as well as in the radial and circumferential directions at the boundary of the hole in flat models. Uniaxial maximum fringe orders were determined and, from these, the biaxial values were obtained by superposition.     

Asymptotic solutions for the Föppl – von Kármán equations governing deflections of thin axisymmetric annular plates

We are concerned with the deformation of thin, flat annular plates under a force applied orthogonally to the plane of the plate. This mechanical process can be described via a radial formulation of the Föppl – von Kármán equations, a set of nonlinear partial differential equations describing the deflections of thin flat plates. We are able to obtain analytical solutions for the radial Föppl – von Kármán equations with boundary conditions relevant for clamped, loosely clamped, and free inner and outer. This permits us to study the qualitative behavior of the out-of-plane deflections as well as the Airy stress function for a number of cases. Provided that an appropriate non-dimensionalization is taken, we find that the perturbation solutions are surprisingly valid for a wide variety of parameters, and compare favorably with numerical simulations in all cases (rather than just for small parameters). The results demonstrate that the ratio of the inner to outer radius of the annular plate will strongly influence the properties of the solutions, as will the specific boundary data considered. For instance, one may choose to fix the plate in place with a specific set of boundary conditions, in order to minimize the out-of-plane deflections. Other boundary conditions may result in undesirable behaviors.     

Surface tension-induced stress concentration around a nanosized hole of arbitrary shape in an elastic half-plane

This paper studies surface tension-induced stress concentration around a nanosized hole of arbitrary shape inside an elastic half-plane. Of particular interest is the maximum hoop stress on the hole’s boundary with relation to the point of maximum curvature and the distance between the hole and the free surface of the half-plane. The shape of the hole is characterized by a conformal mapping which maps the exterior of the hole onto the exterior of the unit circle in the image plane. On using the technique of conformal mapping and analytic continuation, the complex potentials of the half-plane are expressed in a series form with unknown coefficients to be determined by Fourier expansion method. Detailed numerical results are shown for elliptical, triangular, square and rectangular holes. Two basic conclusions are that the hoop stress increases with decreasing hole size and the maximum hoop stress generally appears nearby but not exactly at the point of maximum curvature. In addition, it is shown that the hoop stress nearby the point of maximum curvature on the hole’s boundary increases rapidly with decreasing distance between the hole and the free surface of the half-plane. On the other hand, if the distance between the hole and the free surface is more than three times the hole size, the effect of the free surface on the stress concentration around the hole is ignorable and the elastic half-plane can be treated approximately as an elastic whole plane.     

Photoelastic analysis of stresses around oblique holes

A three-dimensional photoelastic analysis was conducted to determine the magnitude and distribution of stresses around oblique holes in a uniaxially loaded plate. The holes were circular and inclined at angles of 45 deg and 60 deg with the faces of the plate. The plate-thickness-to-hole-diameter ratiot/D was 2.40. One end of each hole was blended to the face of the plate through a break radius equal to the radius of the hole. The plate dimensions were sufficiently large to simulate conditions of an infinite plate. The plates were loaded perpendicular to the plane of skewness. Stress distributions were obtained on sections perpendicular to the direction of loading. Results point to two critical areas of stress concentration: one at the acute-angle intersection of the hole and the surface of the plate and the other in the break-radius area. The stress concentrations in the latter area reach values of 4.6 and 6.7 compared to 3.6 and 4.5 at the acute-angle intersection, for the inclination angles of 45 deg and 60 deg, respectively. A simplified analysis used for the break-radius area gave results in agreement with the experiment. Thus, it was shown that break radii in oblique penetrations may have deleterious rather than beneficial effects. Comparison of results for the acute-angle intersection with existing theoretical and experimental values shows a definite and pronounced dependence of the stress-concentration factor on thickness-to-diameter ratio.     

Magneto-elastic stress induced by an electric current in an infinite thin plate with an elliptical hole

Two-dimensional magnetic field and stress analyses have been presented for soft ferromagnetic, paramagnetic, and diamagnetic materials of an infinite thin plate with an elliptical hole under steady electric current. The magnetic stress has been analyzed in the Maxwell Stress Model. Except for the approximation of the plane stress state since the plate is the thin plate, any assumption is not made for the stress analysis, though the Maxwell stress components are expressed by nonlinear terms. The boundary condition expressed by Maxwell’s stress is completely satisfied without any linear assumptions on the boundary. Two ways for the boundary condition are stated. The analysis of σ _z in the direction of the plate thickness is also carried out. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived, and the magnitude of the stress intensity factor for the magnetic stress and thermal stress due to the Joule heat caused by the electric current is discussed.     

加筋土地基模型试验边界效应与承载性能分析

为进一步探讨边界效应对加筋土地基的影响,基于室内方形基础下加筋土地基大模型试验,采用ABAQUS有限元软件建立加筋土地基数值模型,主要分析了模型宽度L和加载板宽度B对加筋土地基承载性能、地基内部土体应力应变及筋材变形的影响.结果表明:无筋地基与加筋土地基极限承载力均随L/B的减小而增大,当L/B＞5时,可忽略边界效应对...     

A plane theory of inextensible transversely isotropic elastic composites

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic material, with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and traction conditions lead to standard and nonstandard boundary value problems of potential theory. Examples for a rectangular plane, half-plane and infinite plate with elliptic hole, are presented in illustration.     

Large Plastic Deformations Accompanying the Growth of an Elliptical Hole in a Thin Plate

Within the context of plane stress assumptions and approximations, an analytical solution is derived for the finite deformation of a traction-free elliptical hole in an infinite plate with tensile tractions at infinity. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under a deformation theory of plasticity. The governing partial differential equations are parabolic in nature and consequently have a single family of mathematical characteristics or slip lines associated with them. Each particle of mass follows a rectilinear path in the plane defined by its slip line which emanates orthogonally from the elliptical hole. By assuming a constant speed for each particle in the plane, a state of plane equilibrium is realized. The originally elliptical hole expands in the shape of an oval which is a parallel curve to the original ellipse. The slip lines remain orthogonal to the evolving oval hole as a necessary condition for a traction-free interior boundary. This solution also satisfies the material stability criterion that the rate of plastic work be positive throughout the entire body for all time. As this solution has some features associated with large deformation crack problems at elevated temperatures, possible applications include secondary or steady-state creep.     

The concentration of stress and strain in finite thickness elastic plate containing a circular hole

The elastic stress and strain fields of finite thickness large plate containing a hole are systematically investigated using 3D finite element method. It is found that the stress and strain concentration factors of the finite thickness plate are different even if the plate is in elasticity state except at notch root of plate surface. The maximum stress and strain do not always occur on the mid plane of plate. They occur on the mid plane only in thin plate. The maximum stress and strain concentration factors are not on mid plane and the locations of maximum stress and strain concentration factors are different in thick plate. The maximum stress and strain concentration factors of notch root increase from their plane stress value to their peak values, then decrease gradually with increasing thickness and tend to each constant related to Poisson’s ratio of plate, respectively. The stress and strain concentration factors at notch root of plate surface are the same and are the monotonic descent functions of thickness. Their values decrease rapidly and tend to each constant related to Poisson’s ratio with plate thickness increasing. The difference between maximum and surface value of stress concentration factor is a monotonic ascent function of thickness. The thicker the plate is or the larger the Poisson’s ratio is, the larger the difference is. The corresponding difference of strain concentration factor is similar to the one of stress concentration factor. But the difference magnitude of stress concentration factor is larger than that of strain concentration factor in same plate.     

Normal surface displacement around a circular hole by reflection holographic interferometry

In a previous paper¹, a fringe-compensation technique was developed to improve the possibilities of stress analysis by real-time holographic interferometry. The technique is specially well suited for the measurement of small displacements in the direction of viewing. As an application of this method, the surface displacements caused by strains in the thickness direction are measured around a circular hole in a plate loaded in tension in its plane. Independent prior knowledge of the in-plane displacement is required, however, in data processing. An analytical solution to the problem is used for that purpose. The experimental results are compared to those obtained theoretically from the classical two-dimensional analysis, and from a three-dimensional analysis. The two-dimensional theory assumes a state of ‘generalized plane stress’. The three-dimensional theory, made by Alblas², takes into account the existence of stresses in the thickness direction, and the variation of the in-plane stresses through the thickness. Both theories give the same results away from the hole. They differ significantly, however, when the hole boundary is approached, where the proximity of the hole induces three-dimensional effects. The experimentally measured displacement is found to be in good agreement with both theories away from the hole. Close to the hole, a large departure from the two-dimensional results is observed. The experimental results here are close to those of three-dimensional results. The experiment is thus in good agreement with the three-dimensional theory over the whole field. But the two-dimensional theory is valid only at large distances from the hole.     

Bending of a stretched plate containing an eccentrically plate reinforced hole of arbitrary shape

In this paper we consider the problem of a stretched plate containing a hole of arbitrary shape which is reinforced by thickening the plate, on one side only, in a region surrounding the hole. Due to the eccentricity of the reinforcement a bending boundary layer occurs in the neighbourhood of the junction between the plate and the reinforcement. The equations for the moments at the junction are found to be identical to those for the circular hole in Ref. [1]. The boundary layer occurring at a clamped edge of arbitrary shape is also discussed.     

含有非圆形双孔的无限平板中应力的解析解研究

无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.      

光学透视测量玻璃纤维增强复合材料板离面应变场分布

本文通过激光波数扫描干涉测量系统和方法,透视测量商用玻璃纤维增强复合材料板前、后表面的离面应变场分布。实验中分别对有缺陷和无缺陷的商用玻璃纤维增强复合材料样品进行了离面位移场测量和平均轴向正应变场分布计算。无缺陷样品的离面位移场及压缩应变场分布均匀连续,随加载呈现递增压缩变化;有缺陷样品的缺陷周围的离面位移场分布变化无规则,其压缩正应变分布以缺陷孔洞为中心,加载量越大,孔洞周围其压缩正应变值较小的区域越大,压缩正应变集中在远离孔洞的边缘区域,随加载量变化压缩应变场分布无线性规律。实验结果证明,激光波数扫描干涉测量系统和方法准确可靠,它为玻璃纤维增强复合材料板的力学性能测量提供了一个新技术平台。     

Stress analysis of notched anisotropic finite plates under mechanical and hygrothermal loads

Summary Notch-induced stress concentrations in anisotropic composite materials depend on their directional material properties, especially for uniaxially reinforced composites with high-modulus fibres. The design of notched high-performance composites requires therefore a special proof of their notched strength, which includes the structural parameters of the fibre/matrix combination, fibre orientation and layer arrangement. The assessment of the effects of the finite outer boundary is of practical importance when dimensioning critical notched regions. An anisotropic plate with finite dimensions and a hole in its center will be used here to model stress concentrations. The calculation is based on conformal mappings combined with complex-valued stress functions. The outer boundary is described using point-matching and the least-squares method. The solutions are conducive to the assessment of the essential influencing factors of material properties, geometry and loads. Notched finite plates made of fibre/matrix composites, mainly carbon-fibre reinforced polymers, will be presented as illustrations. Received 29 June 1998; accepted for publication 22 October 1998     

Photoelastic stress analysis of an elliptical hole in a thick plate subjected to uniform in-plane compressive Loading

A three-dimensional photoelastic analysis was conducted to determine the stress distribution and concentration around the periphery of a centrally located elliptical hole in a plate of finite thickness. The edge of the plate was subjected to a uniformly distributed compressive uniaxial in-plane load. The principle of superposition was employed to study the effect of uniform biaxial loading.Elliptical holes with five different major/minor axis ratios () ranging from 1.0 to 2.64 were investigated. Among the results of this study, it was established that the variation of the principal stresses at the edge of the hole is not linear across the plate thickness. It was also found that in loading the plate in a direction parallel to the major axis of the ellipse, the value of the maximum tangential principal stress () occurs in a plane other than the middle plane of the plate. However, in loading the plate in a direction either parallel or perpendicular to the major axis, the maximum transverse stress ( _z ) occurs at the middle plane. In addition, the maximum value of ( _z ) was about 20 percent of the maximum value of the tangential stress for all models tested. Furthermore, the effect of the bixial loading has reduced the value of the maximum tangential stress at the periphery of the hole as compared with uniaxial loading.As a three-dimensional theoretical solution does not exist for this problem, the present findings were correlated with the well established two-dimensional solutions.     

形状记忆合金椭圆孔口问题的有限单元法

基于Lagoudas形状记忆合金(SMA)三维本构模型,假设材料为各向同性,推导了SMA平面应力状态的增量型本构方程,继而编写了ABAQUS用户自定义材料(UMAT)子程序,研究了在双向拉伸情况下,外载荷、温度、椭圆孔口长短轴之比对超弹性SMA椭圆孔口板中应力诱发马氏体相变区的影响。数值结果表明:应力诱发马氏体相变首先发生在椭圆孔口长轴端点部位,在外加载荷作用下逐渐扩展到板内,并由内向外形成马氏体相区、相变混合区和奥氏体相区;SMA板内应力诱发马氏体完全相变区面积与施加外载荷成正相关,与温度成负相关;随着椭圆孔口长短轴之比增大,SMA板内应力诱发马氏体完全相变区面积呈现出先减小后增大的趋势;拉应力差值相同时,相较于拉应力沿椭圆孔口长轴方向较大的情况,当拉应力沿椭圆孔口短轴方向较大时,SMA板内完全相变区面积较大,椭圆孔口周边应力集中现象更明显。