Stress measurements in an infinite plate with a hole reinforced by different materials

Stress distributions are measured around the reinforced circular hole in an infinite plate subjected to uniaxial tension. Test plates are made of aluminum and reinforcing rings are of aluminum, brass, copper and mild steel. The relationships between stress-concentration factors, the ratio of Young's moduli of a plate and rings and dimensions of rings are studied. Results are compared with those obtained by the other authors and Gurney's theoretical results.     

Stress concentration around a hole in a radially inhomogeneous plate

The stress concentration factor around a circular hole in an infinite plate subjected to uniform biaxial tension and pure shear is considered. The plate is made of a functionally graded material where both Young’s modulus and Poisson’s ratio vary in the radial direction. For plane stress conditions, the governing differential equation for the stress function is derived and solved. A general form for the stress concentration factor in case of biaxial tension is presented. Using a Frobenius series solution, the stress concentration factor is calculated for pure shear case. The stress concentration factor for uniaxial tension is then obtained by superposition of these two modes. The effect of nonhomogeneous stiffness and varying Poisson’s ratio upon the stress concentration factors are analyzed. A reasonable approximation in the practical range of Young’s modulus is obtained for the stress concentration factor in pure shear loading.     

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].     

Geometric elasticity problem of stress concentration in a plate with a circular hole

We use the geometric elasticity equations [1], which permit relating the medium stress state to the geometry of the Riemannian space generated by the stresses, to consider the plane problem of stress concentration near a circular hole in a thin unbounded plate loaded by normal and tangential stresses. The Riemannian space metric coefficient corresponding to the coordinate normal to the plate plane is treated as the variable thickness of the plate in three-dimensional Euclidean space, which determines the optimal law for the plate material distribution. We consider plates in uniaxial tension, biaxial tension, and shear. For the plate with thickness variation laws thus obtained, we construct direct numerical solutions of the corresponding classical elasticity problems and determine the stress concentration factors.     

On the analysis of neutral holes

When an opening must be located in a tension panel the inevitable stress concentration that results can be reduced if the edge of the opening is reinforced. While there is a variety of solutions for providing reinforcements which reduce stress concentrations in the panel, none of these consider the state of stress that exists in the reinforcement. This paper extends the theory of neutral holes to consider bending of the reinforcement as well as to allow for the use of a different material in the reinforcement. Photoelastic experiments demonstrate the effectiveness of neutral holes for uniform uniaxial tension panels for which the opening consists of parabolic sections.     

Analysis of brittle fracture of foam polystyrene plates with holes

A series of experiments was conducted on brittle fracture of foam polystyrene plates in uniform uniaxial tension and in the presence of stress concentration near slits and holes. Experimental data were analyzed using a numerical algorithm because the ratio between the dimensions of the holes and those of the plates differed from that in known problems for infinite planes and strips with holes. The experimental and calculated data are compared.     

Stress and failure analysis of a glass-epoxy composite plate with a circular hole

Experimental and finite-element analyses are presented for the anisotropic states of stress, strain and fracture of a glass-epoxy plate containing a circular hole and subjected to uniaxial tension. Strains were experimentally measured using foil gages, moiré and birefringent coating. Stresses are computed in the linear range from the measured strains. While the hole reduces the plate strength by a factor of two, the maximum tensile strain at fracture is greater than the ultimate strain in a plate without a hole. Fracture consists of crack initiation at the hole boundary but off the horizontal axis. Away from the hole, failure is accompanied by considerable delamination. Discontinuous crack propagation is present.     

Three-dimensional stress and free vibration analyses of functionally graded plates with circular holes by the use of the graded finite element method

Static and free vibration analyses of plates with circular holes are performed based on the three-dimensional theory of elasticity. The plates are made of a functionally graded material (FGM), and the volume fractions of the constituent materials vary continuously across the plate. The effective properties of the FGM plate are estimated by using the Mori–Tanaka homogenization method. A graded finite element method based on the Rayleigh–Ritz energy formulation is used to solve the problem. Effects of different volume fractions of the materials and hole sizes on the behavior of FGM plates under uniaxial tension are investigated. Natural frequencies of a fully clamped FGM plate with a circular cutout are derived. The results obtained are compared with available experimental data.     

Solution of the plane stress problems of strain-hardening materials described by power-law using the complex pseudo-stress function

In the present paper,the compatibility equation for the plane stress problems of power-law materials is transformed into a biharmonic equation by introducing the so-calledcomplex pseudo-stress function,which makes it possible to solve the elastic-plastic planestress problems of strain hardening materials described by power-law using the complexvariable function method like that in the linear elasticity theory.By using this generalmethod,the close-formed analytical solutions for the stress,strain and displacementcomponents of the plane stress problems’of power-law materials is deduced in the paper,which can also be used to solve the elasto-plastic plane stress problems of strain-hardeningmaterials other than that described by power-law.As an example,the problem of a power-law material infinite plate containing a circular hole under uniaxial tension is solved byusing this method,the results of which are compared with those of a known asymptoticanalytical solution obtained by the perturbation method.     

含孔边非均匀材料圆环的无限大薄板的应力集中系数

对于含圆孔及孔边非均匀材料圆环的无限大薄板，假设非均匀材料的弹性模量沿径向按照指数函数变化，而泊松比为常数，分别导出了双轴拉伸和纯剪切作用时孔边及界面处的应力集中系数的解析解．通过数值算例详细分析了非均匀材料圆环的弹性模量的变化对无限大薄板的孔边及界面处的应力集中系数的影响．研究结果表明，合理选择孔边非均匀材料圆环的材料性能变化参数可有效地缓解薄板的孔边应力集中程度．本文的研究结果可为含圆孔的薄板的设计提供一定的参考．     

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied     

Influence of breadth in bars with reinforced circular holes under tension

This paper reports stress distributions in epoxy-resin bars with reinforced circular holes under tension. The holes were reinforced by bonding aluminum rings to the bars inside the holes. The stress distributions were determined photoelastically. Various proportions of circular reinforcements and different widths of bars were investigated under pure tension. Stress distributions were determined in the epoxy-resin part of the bar, and the maximum shear stresses were given special attention. The relation between maximum shear stress and the width of the bar was determined, and the ranges for which the theoretical solution for an infinite bar approximates that for a finite bar were defined.     

Influences of surface on the interaction between holes or edge

The effects of surface energy on the interaction between holes or edge are investigated. Three typical problems are discussed: (1) an infinite plate containing two holes of unequal size subjected to an all-round tension, (2) a circle disc containing an eccentric hole subjected to uniform pressure on either external or internal surface, (3) a semi-infinite plate containing an unstressed circular hole subjected to a uniform tension parallel to its straight edge. The problems are solved by series expansion in bipolar coordinates. The results show that the surface energy significantly affects the stress concentrations around the holes as the size of the holes shrinks to nanometers. Meanwhile, the interaction between the holes or edge influences the stress distribution around the holes or edge, which becomes evident as the holes or edge close to each other and is affected by the surface effect significantly.     

Biaxial testing of [O2/±45] s graphite/epoxy plates with holes

An experimental investigation was conducted to study the behavior under biaxial-tensile loading of [O₂/±45] _s graphite/epoxy plates with circular holes and to determine the influence of hole diameter on failure. The specimens were 40-cm×40-cm (16-in.×16-in.) graphite/epoxy plates of [O₂/±45] _s layup. Four hole diameters, 2.54 cm (1.00 in.), 1.91 cm (0.75 in.), 1.27 cm (0.50 in.) and 0.64 cm (0.25 in.), were investigated. Deformations and strains were measured using strain gages and birefringent coatings. Biaxial tension in a 2∶1 ratio was applied by means of four whiffle-tree grip linkages and controlled with a servohydraulic system. Stress and strain redistributions occur around the hole at a stress level corresponding to localized failure around the 67.5-deg location and nonlinear strain response at the 0-deg location. Maximum measured strains at failure on the hole boundary are higher (approximately 0.016) than the highest ultimate strain of the unnotched laminate (0.010). Two basic patterns of failure were observed: (a) horizontal cracking initiating at points off the horizontal axis and accompanied by extensive delamination of the subsurface ±45 deg plies, and (b) vertical cracking along vertical tangents to the hole and accompanied by delamination of the outer 0-deg plies. The strength reduction ratios are lower than corresponding values for uniaxial loading by approximately 16 percent, although the stress-concentration factor under biaxial loading is lower.     

Stress-concentration factors at intersecting and closely approaching orthogonal coplanar holes

Using two models, each containing three groups of intersecting or closely approaching holes, 18 different cases were investigated by three-dinensional frozen-stress photoelasticity. In all cases, the coplanar holes were orthogonal to the applied uniaxial tension. Included were three cases of intersecting hoes with square corners and varying hole-diameter ratios. Five cases of intersecting holes with rounded corners were studied for two hole-diameter ratios and varying corner radius. Stresses were determined for two cases where the end of one drilled hole partially penetrated another hole leaving acute corners or feathered edges at the intersection. Three types of closely approaching holes were studied: two cases in which the end of one drilled hole nearly intersected the side of an-other hole, two cases in which the ends of two drilled holes approached each other along a common axis, and four cases (called corner-approach cases) in which the ends of two drilled holes approached each other along orthogonal axes. A stress-concentration factor of 13 was found for one of the partially penetrating drilled-hole cases. The upper limit on stress-concentration factor for a very small hole intersecting a large hole in an infinite body subjected to uniaxial stress is 8.4 for metals. This factor is reduced to 5.2 as the diameters of the intersecting holes become equal. Rounding the corners on intersecting holes reduced the stress concentration by only four to seven percent. Closely approaching drilled holes results in higher stresses than fully intersecting holes when the minimum ligament width is less than ten percent of the hole diameter. Corner approach cases do not result in high stress-concentration factors as long as the ligament width is greater than the percent of the hole diameter.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

Behavior of Pin-loaded Laminated Composites

In this study, an investigation was carried out to determine the effects of joint geometry and fiber orientation on the failure strength and failure mode in a pinned joint laminated composite plate. Behavior of pin-loaded laminated composites with different stacking sequence and different dimensions has been observed experimentally. E/glass–epoxy composites were manufactured to fabricate the specimens. Mechanical properties of the composites were characterized under tension, compression and in-plane shear in static loading conditions. Laminated composites were loaded through pins. Single-hole pin-loaded specimens were tested for their tensile response and width-to-hole diameter (W/D) and edge distance-to-hole diameter (E/D) ratios evaluated. A series of experiments was performed with six different material configurations ([0/±45]_s–[90/±45]_s, [0/90/0]_s–[90/0/90]_s and [90/0]_2s–[±45]_2s), in all, over 120 specimens. E/D ratios and W/D ratios of plates were changed from 1 to 5 and 2 to 5, respectively. Failure propagation and failure type were observed on the specimens. The influence of the joint geometry on the strength of the pin-loaded composites was assessed. When laminated composite plates were loaded to final failure, three basic failure modes consisting of net-tension, shear out and bearing failure were observed for the different geometric dimensions. All the connections tested showed that the fiber orientations have a definite influence on the position around hole circumference at which failure initiated. Net-tension failure occurred for specimens that had small width and large end distance. When the width was increased, the specimens which had small end distances failed in the shear-out modes. When the end distance was increased, bearing failure developed in addition to shear-out failure. The experimental results showed that the ultimate load capacities of E/glass–epoxy laminate plates with pin connection were increased by increasing W and E. However, increasing the E/D and W/D ratios beyond a critical value has an insignificant effect on the ultimate load capacity of the connection.     

Stress in elastic plates reinforced by fibres lying in concentric circles

We Consider fibre-reinforced elastic plates with the reinforcement continuously distributed in concentric circles ; such a material is locally transversely isotropic, with the circumferential direction as the preferred direction. For an annulus bounded by concentric circles, the exact solution of the traction boundary value problem is obtained. When the extension modulus in the fibre direction is large compared to other extension and shear moduli, the material is strongly anisotropic. For this case a simpler approximate solution is obtained by treating the material as inextensible in the fibre direction. It is shown that the exact solution reduces to the inextensible solution in the appropriate limit. The inextensible theory predicts the occurrence of stress concentration layers in which the direct stress is infinite ; for materials with small but finite extensibility these layers correspond to thin regions of high stress and high stress gradient. A boundary layer theory is developed for these regions. For a typical carbon fibre-resin composite, the combined boundary layer and inextensible solutions give an excellent approximation to the exact solution. The theory is applied to the problem of an isotropic plate, under uniform stress at infinity, containing a circular hole which is strengthened by the addition of an annulus of fibre-reinforced material.     

正交各向异性板孔销接触问题的复变函数解法

本文提出一个计及轴销变形与接触区摩擦效应的弹性轴销模型,并采用弹性理论的复变函数解法,研究了正交各向异性无限大板的孔销接触问题.分析表明,板件的孔边应力分布与板件、轴销的材料性能密切相关,而且,摩擦系数的大小对应力分布也存在明显影响.     

Gradient model in the problem of stress-concentration around a circular hole in two-component stochastic composites

The gradient model of stochastically inhomogeneous media is used to study the stress concentration around a circular hole in a two-component elastic composite. The study is based on a general solution of the system of equilibrium equations expressed in terms of harmonic functions and functions that satisfy the Helmholtz equation. This solution is used to solve problems for an infinite plane with a circular hole under uniform and uniaxial tension. The results obtained are compared with the solutions found using the theory of effective moduli, which is simpler __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 41–53, December 2007.