Uniform Normal Loading Research Articles

Semi-inverse method for solving circular arc crack problems

A new approach to a certain class of circular arc crack problems is proposed in this paper. The problems are reduced to the problem of linear relationship of the boundary value (the Hilbert problem). By means of the harmonic measure and conformal mapping technique, the closed solutions are given for a plane with a single circular arc crack subjected to the uniform normal and tangential loads on the part near one end of the crack and for the same plane under concentrated normal and tangential loads at an arbitrary point on the arc crack surface. The stress intensity factors at the crack tips under consideration are determined. Some known results for the plane with the central straight crack are shown as the limiting cases of the present solutions.

An Asymptotic Solution for Non-Linear Behavior of Imperfect Shallow Spherical Shells

AbstractThis paper deals with non-linear behavior of an elastically restrained shallow spherical shell with imperfections and subjected to uniform normal load by the asymptotic iteration method. An asymptotic relation between the external load and central transverse displacement (deflection) of the structure incorporating the effects of the characteristic geometrical parameter, imperfection factor and edge-restraint coefficients is formulated. The corresponding solutions resulting from the use of the second and third iteration processes are compared with each other and available data. The numerical calculations show that the second iteration is adequate for the non-linear analysis of the structure. The effects of geometrical imperfections and boundary constraint coefficients on buckling of the structures are presented graphically.

Transverse Stress Decay in a Specially Orthotropic Strip under Localized Normal Edge Loading

Solutions are presented for the stresses in a specially orthotropic infinite strip which is subjected to localized uniform normal loading on one edge while the other edge is either restrained against normal displacement only, or completely fixed. The solutions are used to investigate the diffusion of load into the strip and in particular the decay of normal stress across the width of the strip. For orthotropic strips representative of a broad range of balanced and symmetric angle-ply composite laminates, minimum strip widths are found that ensure at least 90% decay of the normal stress across the strip. In addition, in a few cases where, on the fixed edge the peak shear stress exceeds the normal stress in magnitude, minimum strip widths that ensure 90% decay of both stresses are found. To help in putting these results into perspective, and to illustrate the influence of material properties on load diffusion in orthotropic materials, closed-form solutions for the stresses in similarly loaded orthotropic half-planes are obtained. These solutions are used to generate illustrative stress contour plots for several representative laminates. Among the laminates, those composed of intermediate-angle plies, i.e., from about 30° to 60°, exhibit marked changes in normal stress contour shape with stress level. The stress contours are also used to find 90% decay distances in the half-planes. In all cases, the minimum strip widths for 90% decay of the normal stress exceed the 90% decay distances in the corresponding half-planes, in amounts ranging from only a few percent to about 50% of the half-plane decay distances. The 90% decay distances depend on both material properties and the boundary conditions on the supported edge.

Transverse Stress Decay in a Specially Orthotropic Strip under Localized Normal Edge Loading

Solutions are presented for the stresses in a specially orthotropic infinite strip which is subjected to localized uniform normal loading on one edge while the other edge is either restrained against normal displacement only, or completely fixed. The solutions are used to investigate the diffusion of load into the strip and in particular the decay of normal stress across the width of the strip. For orthotropic strips representative of a broad range of balanced and symmetric angle-ply composite laminates, minimum strip widths are found that ensure at least 90% decay of the normal stress across the strip. In addition, in a few cases where, on the fixed edge the peak shear stress exceeds the normal stress in magnitude, minimum strip widths that ensure 90% decay of both stresses are found. To help in putting these results into perspective, and to illustrate the influence of material properties on load diffusion in orthotropic materials, closed-form solutions for the stresses in similarly loaded orthotropic half-planes are obtained. These solutions are used to generate illustrative stress contour plots for several representative laminates. Among the laminates, those composed of intermediate-angle plies, i.e., from about 30 degrees to 60 degrees, exhibit marked changes in normal stress contour shape with stress level. The stress contours are also used to find 90% decay distances in the half-planes. In all cases, the minimum strip widths for 90% decay of the normal stress exceed the 90% decay distances in the corresponding half-planes, in amounts ranging from only a few percent to about 50% of the half-plane decay distances. The 90% decay distances depend on both material properties and the boundary conditions on the supported edge.

Finite element analysis of the contact stresses in diamond coatings subjected to a uniform normal load

Abstract Finite element analysis is used to predict the stress distribution in a diamond-coated substrate. The model examines the effect of coating thickness in detail, and compares and validates the finite element predictions with experimental results obtained from the soft impressor method applied to a series of diamond coatings on a silicon substrate. The results demonstrate that the performance of a coating can be related to the ratio of coating thickness to contact radius ( t / a ), so for example, for the silicon substrate most of the protection is achieved when the t / a ratio is about 0.1, and for the diamond coating there is no advantage in using coatings with t / a ratios grater than 1.3.

Three-dimensional exact solution for inhomogeneous and laminated piezoelectric plates

This paper presents an exact three-dimensional method of solution for the distribution of mechanical and electric quantities in interior points of a symmetrically inhomogeneous and laminated piezoelectric plate in the framework of linear theory of piezoelectricity. A transfer matrix method and asymptotic expansions are used as the elements of the formulation. The full three-dimensional electroelasticity solution is generated from a set of two-dimensional differential equations on the midplane which, when degenerated to pure elasticity, are found to be the same as those for an equivalent classical elastic plate model. As an illustrative example, the analysis considers a rigidly clamped, elliptic monoclinic piezoelectric plate which is inhomogeneous through the thickness but symmetric with respect to the midplane and under uniform normal loading both mechanically and electrically. Excluding boundary layer effects, an exact closed-form analytical solution for the fully electromechanically coupled problem is developed as a consequence of the asymptotic expansion's termination after a few terms. Apart from some important properties already noted for a purely elastic plate in the same configuration, this exact solution reveals that certain significant electric quantities have been unjustifiably oversimplified by some of the existing piezoelectric plate models.

Hybrid changeable basis galerkin technique for nonlinear analysis of structures

Based on the asymptotical perturbation method and the Galerkin technique, the hybrid changeable basis Galerkin technique is presented for predicting the nonlinear response of structures. By the idea of changeable basis functions first proposed, it greatly reduces calculation and is easily used in other numerical discretization techniques, such as finite element method etc. It appears to have high potential for solution of nonlinear structural problems. Finally, the effectiveness of this technique is demostrated by means of two numerical examples: the large deflection of circular plates objected to uniform normal load and the large deflection of spherical caps under centrally distributed pressures.

Exact three-dimensional elasticity solutions for bending of moderately thick inhomogeneous and laminated strips under normal pressure

An exact three-dimensional solution is presented for the deformation and stress distribution in a semi-infinite strip clamped along its two edges, and subjected to uniform normal loading of the lateral surfaces. The strip is of constant moderate thickness and composed of anisotropic elastic material which is arbitrarily inhomogeneous in the through-thickness direction. The only material symmetry assumed is that of reflectional elastic symmetry in planes parallel to the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise-constant properties through the thickness.The general method of solution is to reformulate the full three-dimensional elasticity equations in a way that reduces the problem to solving a system of partial differential equations in the two in-plane independent variables only, and then obtaining asymptotic solutions in terms of an aspect ratio of the thickness divided by a typical in-plane length. In general, successive terms are expressed in terms of the approximate “classical laminate” solution. In the present problem, this solution is very simple and the expansion terminates after no more than three terms. This gives a closed-form analytical solution that is valid for any aspect ratio.

A contact problem with friction and adhesion for an elastic layer with stiffeners

The contact of an elastic layer with an infinite stiffener to which a uniform constant normal load and a concentrated tangential force are applied, is considered. In the neighbourhood of the point of application of this force, on the line of the contact of the stiffener with the layer a segment is separated out, on which the effect of the Coulomb friction is taken into account. Outside this segment the stiffener and the layer are under conditions of complete adhesion. The problem is reduced to a Prandtl-type integro-differential equation specified on two semi-infinite segments, for whose solution an analytical method is proposed. The method is based on reducing the equation to a vectorial Riemann problem and then to an algebraic Poincaré-Koch system. The latter admits of an explicit solution and also inversion through recurrent relations that are effective when using numerical computations. The length of the Coulomb friction zone and the contact tangential stresses in the adhesion zone are determined. Unlike Melan's problem [1] the contact stresses have no logarithmic singularity and are continuous everywhere in the contact area. The solution of the problem of the contact of a layer with a finite stiffener subject to a uniform pressure along the whole length and to an extension by forces concentrated at the tips is also obtained. The contact area is divided into an intermediate zone of adhesion and two zones of coulomb friction. The problem is reduced to a Prandtl-type integro-differential equation specified on the segment, and it is solved by analogy with the solution of the equation of the first problem. Such a formulation of the problem implies that the contact tangential stresses are bounded at the tips of the stiffener and are continuous at the points of the boundary between the zones of adhesion and Coulomb friction. When adhesion occurs along the entire line of contact the tangential stresses, in general, have a root singularity [2]. In the problem of the contact of a plane punch with a half-plane under conditions of friction and adhesion, the contact stresses at the tips of the punch have [3] a power singularity (that differs from a root one).

On plastic zones and fracture strengths in some metal matrix composites

Based on a Dugdale-type approach for cancelling the singularity at the crack tip, an attempt was made to predict the overall stress distributions, as well as the size of the plastic zone found in notched metal matrix composite plate. The work was developed by considering notched, unidirectionally reinforced fibrous metal matrix composite plate under uniform normal tension load. Predictions of the proposed method were compared with the experimental results, and a fairly good agreement was observed. Simple closed form expressions for the local stresses and fracture strength are provided.

An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness

An exact three-dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces, and clamped along its edge. The plate is assumed to be of constant, moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through-thickness direction but symmetric about the mid-plane. The only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid-plane. A transfer matrix method is used which, without making any further assumptions, gives the exact solution at each point in the plate in terms of the stress and displacement at the mid-plane. The two-dimensional differential equations governing these mid-plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through-thickness weighted averages of the inhomogeneous moduli. The solution of the two-dimensional problem is known for such a plate when subject to the specified surface and edge conditions, and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three-dimensional elasticity problem, with edge displacement conditions satisfied on the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness.

Large deflexions of a uniformly loaded, elastically restrained composite strip

An analysis is given of the large-deflexion behaviour under uniform normal load of a multi-layered composite strip that exhibits coupling between moments and planar strains and whose planar boundary displacements are elastically restrained. The strip forms part of a uniformly loaded infinite plate supported on a grid of stringers that is itself supported on an orthogonal grid of ribs.

Stress Systems in Orthotropic Quarter Plane

A solution is derived for stresses in an orthotropic quarter plane loaded by (1) A tangential concentrated line load at distance a from the edge; (2) a uniform normal load over length a and at distance a from the edge; and (3) a uniform normal load over length a at the edge. The results may find application in reinforced‐earth and geotextile retaining walls, vertical slopes of horizontally jointed rock, and unsupported excavations of anisotropic soil.

A parametric study of the axisymmetric full-range response of thickness-tapered circular steel plates

The incremental equations governing the full-range axisymmetric flexural response of thickness-tapered circular plates and their numerical solution via a finite-difference implementation of the Dynamic Relaxation (DR) technique are briefly outlined. A computer implementation of the DR analysis is used to carry out a parametric study of the response of thickness-tapered plates subjected to uniform normal pressure loading. Dimensionless results (deflections etc.) spanning the practical range of linear thickness tapers are presented for simply supported and clamped, slender and stocky plates.

Transient Stress of an Elastic Half Space Subjected to a Uniform Impulsive Load in a Rectangular Region of Its Surface

This paper presents a solution of transient stress in the interior of an elastic half space under a uniform normal or tangential impulsive load acting on a semi-infinite strip and a rectangular region of its surface. First, the response of a half space loaded over a semi-infinite strip, finite in width is obtained by superimposing the solution which has been derived to the problem of one quarter. Subsequently, the solution to a half space loaded on a rectangular region is obtained by the superposition of response of the semi-infinite strip. The response of one quarter pressure may be derived by using double Fourier transforms on space, Laplace transform on time and Cagniard's method. The solution has singularities on the real axis of Fourier transform plane. We must take notice of the problem of Whether the singularities are inside the closed curve of Cagniard's path in the super position or not.

The analysis of deformations caused by loading applied to the walls of a circular tunnel

AbstractAn analytical solution to the problem of a load applied to a portion of the wall of a circular tunnel installed deep in a homogeneous isotropic material has been found. The solution is obtained by taking a Fourier transform, in the axial direction, of the field quantities and then representing these as a Fourier series in the polar angle. The integration which is necessary for the inversion of Fourier transforms has to be performed numerically using Gaussian quadrature.The solution has been used to analyse the problem of uniform normal loading applied over rectangular patches of the tunnel wall. An indication of how the solution can be applied to the tunnel jacking problem is given.

Interfacial cracks between two dissimilar anisotropic materials

The two-dimensional problem of a crack opened in a dissimilar anisotropic composite under uniform normal loading is considered. In both the phases, the stresses and displacements have been formulated with the stress potentials of Muskhelishvili. The corresponding boundary value problem may be reduced to a pair of dual-homogeneous Hilbert equations. The order of signularity of the stresses at the crack tips has been considered.

Collapse of Thin Orthotropic Elliptical Cylindrical Shells Under Combined Bending and Pressure Loads

The elastic collapse of thin orthotropic elliptical cylindrical shells subject to pure bending alone or combined bending and uniform normal pressure loads has been studied. Nonlinear finite deflection thin shell theory is employed and this reduces the problem to a set of nonlinear ordinary differential equations. The resulting two-point nonlinear boundary-value problem is then linearized, using quasi-linearization, and solved numerically by the “shooting technique.” Some experimental work has been carried out and the results are compared with the theoretical predictions.

Large deformation possible in every isotropic elastic membrane

This paper is concerned with static solutions of finitely deformed elastic membranes regarded as thin shells. It deals with deformations that can be maintained, in the absence of body force, in every isotropic elastic membrane by the application of edge loads and/or uniform normal surface loads on the major surfaces of the thin shell-like body. The solutions, which are valid for both compressible and incompressible materials, are obtained with the use of a strain energy response function which depends on the metric tensor of the membrane in its deformed configuration. The main results are summarized by several theorems and their corollaries in accordance with three mutually exclusive cases for which the initial undeformed surface of the membrane (which may be a sector of a complete or closed surface) is, respectively, developable, spherical and a surface of variable Gaussian curvature satisfying certain differential criteria. The corresponding deformed surfaces are, respectively, a plane or a right circular cylinder, a sphere and a surface of constant mean curvature. These results are exhaustive in that they represent all finite deformation solutions possible in every isotropic elastic material characterized by the strain energy response mentioned above. Also discussed are some special cases of the general results and several families of solutions in terms of an alternative description which should be useful in application and which permit easy interpretations.

Teilweise unverbundene Trennfl�chen zwischen K�rpern aus verschiedenen Werkstoffen bei Normalkraft-und Schubbelastung

The stress distribution in isotropic half-spaces which are bonded together at their plane interface except over a region of finite length and submitted to uniform normal and shear loading along this finite crack was determined. The problem was formulated by setting the admissible equations for stresses and displacements in both phases. It was shown that the corresponding boundary-value problem may be reduced to a Hilbert problem and its solution may be readily determined. The expressions for the complex functions of the Muskhelishvili formulations were used to study the type and order of singularities at the crack tips by using the method of caustics [1].