Repair of a Plate with a Circular Hole by Applying a Patch

The stressed state of a thin elastic infinite plate with a circular hole covered by a circular patch of a greater radius is considered. The center of the hole coincides with the center of the patch. The patch is attached to the plate along its entire boundary. Stresses are prescribed at infinity on the plate and at the hole boundary. Complex Muskhelishvili potentials are found by the method of power series, and the behavior of stresses on the patch–plate interface and at the hole boundary is studied.     

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.     

Scattering of flexural wave in a thin plate with multiple circular holes by using the multipole Trefftz method

The scattering of flexural wave by multiple circular holes in an infinite thin plate is analytically solved by using the multipole Trefftz method. The dynamic moment concentration factor (DMCF) along the edge of circular holes is determined. Based on the addition theorem, the solution of the field represented by multiple coordinate systems centered at each circle can be transformed into one coordinate system centered at one circle, where the boundary conditions are given. In this way, a coupled infinite system of simultaneous linear algebraic equations is derived as an analytical model for the scattering of flexural wave by multiple holes in an infinite plate subject to the incident flexural wave. The formulation is general and is easily applicable to dealing with the problem containing multiple circular holes. Although the number of hole is not limited in our proposed method, the numerical results of an infinite plate with three circular holes are presented in the truncated finite system. The effects of both incident wave number and the central distance among circular holes on the DMCF are investigated. Numerical results show that the DMCF of three holes is larger than that of one, when the space among holes is small and meanwhile the specified direction of incident wave is subjected to the plate.     

The plane problem of an elliptically reinforced circular hole in an anisotropic plate or laminate

Summary  Within the scope of linear elasticity, the in-plane problem of an anisotropic plate or laminate with a circular hole and an elliptical hole reinforcement is considered. Arbitrary anisotropic elastic stiffnesses are allowed for the base plate and the reinforcement material, and for the reinforcement there is no restriction for its elliptical shape and size. The analysis of the problem is performed by the complex potential method with appropriately chosen series representations inside and outside the reinforcement region. The derived closed-form solution provides all resultant in-plane stresses and deformations within and around the hole reinforcement with little computational effort and at high accuracy. The determined solution allows a proper and effective assessment of hole reinforcements for many technical applications. Received 26 June 2000; accepted for publication 26 September 2000     

Elastoplastic state of flexible cylindrical shells with two circular holes

The elastoplastic state of thin cylindrical shells with two equal circular holes is analyzed with allowance made for finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distribution of stresses along the hole boundary and in the stress concentration zone (when holes are closely spaced) is analyzed by solving doubly nonlinear boundary-value problems. The results obtained are compared with the solutions that allow either for physical nonlinearity (plastic strains) or geometrical nonlinearity (finite deflections) and with the numerical solution of the linearly elastic problem. The stresses near the holes are analyzed for different distances between the holes and nonlinear factors.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 107–112, October 2004.     

Hole-shape optimization in a finite plate in the presence of auxiliary holes

Minimizing the stress concentration around holes in uniaxially loaded finite plates is an important consideration in engineering design. One method for reducing the stress concentration around a central circular hole in a uniaxially loaded plate is to introduce smaller auxiliary holes on either side of the original hole to help smooth the flow of the tensile principal-stress trajectories past the original hole. This method has been demonstrated by Heywood and systematically studied by Erickson and Riley. Erickson and Riley show that for a central-hole diameter-to-plate width ratio of 0.222, the maximum stress reduction is up to 16 percent. In recent work, Durelliet al. show that the stress concentrations around holes in uniaxially loaded plates can be minimized by changing the hole shape itself till an optimum hole profile with constant stress values respectively on the tensile and compressive segments of the hole boundary is reached. By this technique the maximum stress reduction obtained for the above case is up to 20 percent. In the present work, starting with the optimum sizes and locations of central and auxiliary circular holes for a finite plate given by Erickson and Riley, a systematic study of the hole-shape optimization is undertaken. A two-dimensional photoelastic method is used. For a central-hole diameter-to-plate width ratio of 0.222, the reduction in stress-concentration factor obtained after hole-shape optimization is about 30 percent. It is also shown that it is possible to introduce the ‘equivalent ellipse’ concept for optimized holes.     

Fatigue growth modeling of cracks emanating from a circular hole in infinite plate

This paper presents a numerical approach of fatigue growth analysis of cracks emanating from a hole in infinite elastic plate subjected to remote loads. It involves a generation of Bueckner’s principle and a hybrid displacement discontinuity method (a boundary element method) proposed recently by the senior author of the paper. Because of an intrinsic feature of the boundary element method, a general crack growth problem can be solved in a single region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is modeled conveniently by adding new boundary elements on the incremental crack extension to the previous crack boundaries. As an example, fatigue growth process of an inclined crack in an infinite plate under uniaxial cycle load is modeled to illustrate the effectiveness of the numerical approach. In addition, fatigue growth of cracks emanating from a circular hole in infinite elastic plate subjected to remote loads is investigated by using the numerical approach. Many numerical results are given     

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.     

Applied alternating method to analyze the stress concentration around interacting multiple circular holes in an infinite domain

This paper presents an efficient alternating method for analyzing the interactions among multiple circular holes in a two-dimensional infinite domain. An analytical solution is derived for a single circular hole in an infinite domain subjected to the arbitrary tractions across the circle boundary to achieve this purpose. This analytical solution correlates with a successive iterative superposition process capable of satisfying the prescribed boundary condition for each circular hole of the problem. In addition, several perforated plate problems are solved to demonstrate the proposed methods validity. The computed results and the available referenced solutions closely corresponds to each other and indicates the methods accuracy and efficiency.     

Strain and fracture of circular plates under static and dynamical loading by a spherical body

The strain and fracture of plates under the action of a load normal to their planes was studied in numerous papers. A review of publications in this field in the case of impact by a freely flying body is given in [1–3]. At first, researchers’ attention was mainly paid to the so-called ideal version of collision in which the normal impact of a rigid body on the plate center was considered and the boundary conditions did not affect the results of impact. The plate strains were studied near and in the region of impact, the minimal velocities were determined for a body of some specific shape for which the plate is punched through (the so-called ballistic limit); the shapes of fractured punched plates and the residual velocity of the body if its initial velocity exceeds the ballistic limit were also determined. In the last years, the more complicated cases of collision have been studied, namely, the case in which the impact is not directed along the normal to the plate plane and the impact velocity vector does not coincide with the body symmetry axis as well as the case of impacts on shells. The research in this field was represented in [2, 3]. But in this case the influence of the boundary conditions is still considered insufficiently. This gap was indicated in [2].In the present paper, we study the normal impacts of spherical bodies and deformable cylindrical bodies with spherical heads on circular plates for various boundary conditions and mechanical characteristics of their material. We consider the plate strains, determine the impact velocity at which the plate is punched through, and clarify the mechanism and the sequence of the plate fracture and break-though depending on their mechanical characteristics and boundary conditions. We make an attempt to perform numerical studies of the dynamic deflection at the center of a plate fixed on the boundary using its experimentally determined quasistatic rigidity and taking into account the boundary conditions for determining the associated mass. We estimate the influence of the body mass on the ballistic limit. The use of rigid spherical bodies permits treating any variations in the results of impacts as a characteristic reaction of the plates themselves, because in this case it is unnecessary to deal with the body orientation with respect to the velocity vector. For impacts with such bodies, we used plates made of aluminum alloys and of lead. We studied how the strength of cylindrical bodies with spherical heads made of plasticine or lead affects the strain of plates made of AMTsM alloy.     

Contact problems for a circular plate with sliding fixation at the end face

Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained illposed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9].     

快速多极虚边界元法对含圆孔薄板有效弹性模量的模拟分析

针对虚边界元法,引入快速多极展开和广义极小残值法(GMRES)的思想,以形成快速多极虚边界元法的求解思想,并将此方法用于含圆孔薄板有效弹性模量的模拟分析.由于本文方法采用了"源点"多极展开和"场点"局部展开的组合处理方案,从而使得原问题方程组求解的计算耗时量和储存量降至与所求问题的计算自由度数成线性比例.本文工作的研究目的在于:提高虚边界元法在普通台式机上的运算能力和拓宽虚边界元法对大规模复杂问题的求解(或数值模拟).文中给出了均布圆孔的正方形薄板和之字形分布圆孔薄板二个算例,以验证该方法的可行性,计算精度和计算效率.     

Analysis of the stress-strain state of an anisotropic plate with an elliptic hole and thin elastic inclusions

We solve the problem of determining the stress-strain state of an anisotropic plate with an elliptic hole and a system of thin rectilinear elastic inclusions. We assume that there is a perfect mechanical contact between the inclusions and the plate. We deal with a more precise junction model with the flexural rigidity of inclusions taken into account. (The tangential and normal stresses, as well as the derivatives of the displacements, experience a jump across the line of contact.) The solution of the problem is constructed in the form of complex potentials automatically satisfying the boundary conditions on the contour of the elliptic hole and at infinity. The problem is reduced to a system of singular integral equations, which is solved numerically. We study the influence of the rigidity and geometry parameters of the elastic inclusions on the stress distribution and value on the contour of the hole in the plate. We also compare the numerical results obtained here with the known data.     

Uniaxial tension of a plate with a circular hole at finite strain

Some kinematic characteristics of a geometrically nonlinear deformation process obtained experimentally at uniaxial tension of a plate with a circular hole are given. The plate is made of a highly elastic material. The geometric linearization method is used to determine the numerical characteristics of the deformation process under study.     

Bending of a highly stretched plate containing an eccentrically plate-reinforced circular hole

In this paper, we consider the problem of finding the stress distribution in a highly stretched plate containing a circular hole that is eccentrically reinforced by thickening the plate, on one side only, in an annular region concentric with the hole. A solution of the nonlinear Kármán plate equations is obtained that is asymptotically valid for large membrane stresses. We show that, except for a narrow bending boundary layer in the neighbourhood of the boundary between the reinforced area and the rest of the plate, a state of plane stress prevails and the reinforced area undergoes a transverse deflection that brings its middle surface into the plane of the middle surface of the plate.     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

The solution of a crack emanating from an arbitrary hole by boundary collocation method

In this paper a group of stress functions has been proposed for the calculation of a crack emanating from a hole with different shape (including circular, elliptical, rectangular, or rhombic hole) by boundary collocation method. The calculation results show that they coincide very well with the existing solutions by other methods for a circular or elliptical hole with a crack in an infinite plate. At the smae time, a series of results for different holes in a finite plate has also been obtained in this paper. The proposed functions and calculation procedure can be used for a plate of a crack emanating from an arbitrary hole.     

Tensionless contact of a finite circular plate

A general formulation is developed for the contact behavior of a finite circular plate with a tensionless elastic foundation. The gap distance between the plate and elastic foundation is incorporated as an important parameter. Unlike the previous models with zero gap distance and large/infinite plate radius, which assumes the lift-off/separation of a flexural plate from its supporting elastic foundation, this study shows that lift-off may not occur. The results show how the contact area varies with the plate radius, boundary conditions and gap distance. When the plate radius becomes large enough and the gap distance is reduced to zero, the converged contact radius close to the previous ones is obtained.     

Flexural wave scattering in a quarter-infinite thin plate with circular scatterers

Scattering of flexural waves by circular scatterers in a quarter-infinite thin plate is formulated using the wave expansion method together with the method of images. The scattered waves are expressed as a summation series of wave functions and the unknown scattering coefficients are determined by enforcing boundary conditions at the scatterers. Both holes and rigid scatterers are studied. Simply-supported and roller-supported boundary conditions on the quarter-infinite thin plate are also considered. The analysis can be used to determine the stress concentration caused by circular scatterers in quarter-infinite thin plates.     

Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading

Summary Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems are investigated by means of a so-called deformation map. The deformation map was further used for stability considerations of geometrically nonlinear shells, see Shilkrut [1, 2]. The map reveals the complete picture of the axially symmetric deformations and the stability of the investigated structure. The equilibrium differential equations for the above mentioned circular plate were derived by Timoshenko [3]. The boundary value problem of the investigated structure is transformed to an initial value problem (Cauchy's problem). Then the Runge-Kutta (R. K.) method can be used to solve numerically the equilibrium equations. The geometrically nonlinear, simply supported circular plate subjected to uniform radial force and uniform radial bending moment acting along the supported edge is investigated as example, and some new qualitative and quantitative results are obtained. This approach can be used without essential difficulties for the investigation of axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems in elastic and non-elastic fields.

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Axialsymmetrische Verformung und Stabilität geometrisch nichtlinearer Kreisplatten unter mehrparametrischer statischer Belastung

Übersicht Zur Untersuchung axialsymmetrischer, geometrisch nichtlinearer Verformung von Kreisplatten und ihrer Stabilität bei mehrparametrischer Belastung wird eine sog. Deformationskarte benutzt. Sie wurde auch für Stabilitätsbetrachtungen geometrisch nichtlinearer Schalen benutzt, s. Shilkrut [1,2]. Die Karte zeigt das vollständige Bild der axialsymmetrischen Verformung und die Stabilität der untersuchten Struktur. Das Randwertproblem zu den differentiellen Gleichgewichtsbedingungen, die für die betrachtete Platte von Timoshenko [3] hergeleitet wurden, wird in ein Anfangswertproblem (Caudy-Problem) überführt, welches numerisch nach der Methode von Runge-Kutta gelöst wird. Als Beispiel wird die nichtlineare Kreisplatte unter radialer Zug-und Biegemomentenbelastung am einfach gestützten Umfang untersucht, und man erhält einige neue qualitative und quantitative Ergebnisse. Die Methode läßt sich ohne wesentliche Schwierigkeiten auch auf axialsymmetrische, nichtlineare Verformungen und die Stabilität von Kreisplatten unter anderen mehrparametrischen statischen Belastungen im elastischen und nichtelastischen Bereich anwenden.