Stress concentration factors due to the bending of a thick plate with circular hole

Summary Solutions of a biharmonic equation together with four Helmholtz equations governing a twelfthorder thick plate theory in cylindrical coordinates are derived in a form suitable for finding corrections to previous lower-order thick plate results. As an example, for an infinite plate with a circular hole subjected to bending moments at two parallel outside edges, detailed comparisons of values of a stress couple concentration factor and a stress concentration factor are made between the present theory and previous results due to Alblas, Reissner, and Cheng.

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Spannungskonzentrationsfaktoren für die Biegung dicker Platten mit Kreisloch

Übersicht Für die Grundgleichungen einer Theorie 12. Ordnung für dicke Platten, bestehend aus einer biharmonischen Gleichung und vier Helmholtz-Gleichungen, werden in Polarkoordinaten Lösungen bestimmt, um Korrekturen zu früheren Ergebnissen nach Theorien geringerer Ordnung angeben zu können. Als Beispiel werden für eine unendliche, an zwei parallelen Seiten durch ein Biegemoment belastete Platte mit Kreisloch die Konzentrationsfaktoren für Biegemoment und Randspannung verglichen, die sich nach der hier benutzten Theorie und aus früheren Arbeiten von Alblas, Reissner und Cheng ergeben.

     

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].     

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.     

Unbonded contact between a circular plate and an elastic half-space

The paper concerns the unbonded contact between a thin circular plate of finite radius, governed by Kirchhof or Reissner theory, pressed by means of rotationally symmetric distributed load and its own weight against the surface of an elastic half-space. The contact is assumed frictionless and unbonded. A Hankel transform solution is used for the half-space and the plate deflection is found by inverting the plate equation. The coefficients in a power expansion are obtained by equating plate and half-space deflections at a number of points in the contact region. The variation of contact radius with plate radius, the radius of the uniformly applied load, and the relative stiffness of plate and foundation, is displayed in a series of figures.     

Influence of elastic properties on the stress state of a nonthin transversely isotropic plate with a circular hole

The paper outlines a method for constructing the general analytic solution to the system of equilibrium equations of nonthin transversely isotropic plates. The method uses the Fourier–Legendre series expansion of the unknown functions with respect to the thickness coordinate. The stress state near a circular hole in a nonthin plate subject to tensile and shear stresses at infinity is analyzed     

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.     

Thermal impact on a circular sandwich plate on an elastic foundation

Symmetric transverse vibrations of a circular metal-polymer sandwich plate under a thermal impact are studied. The plate is connected with an inertialess Winkler foundation. The facesheets are assumed to satisfy the Kirchhoff hypothesis and the deformed normal in the low-density core is rectilinear and incompressible across thickness. Analytical solutions are obtained and their numerical analysis is given.     

Extension of a viscoplastic plate with a circular hole

International Applied Mechanics -     

Displacement field around a circular hole in a viscoelastic plate

Moiré experimental techniques are used to measure displacement fields in viscoelastic plates undergoing large deformations at elevated temperatures. These experimental procedures are applicable to determining displacement fields in nonlinear materials. As preliminary information, the material properties are determined from creep studies. The moiré method is used to determine the strains under constant load and isothermal conditions. Tests are conducted for several combinations of load and temperature for 2.5 decades of time. Assuming thermorheologically simple behavior, the data are shifted to establish the creep extensional compliance over ten decades in time. The constitutive equations are formulated as integral equations, the kernels of which are the functions that were measured in this work. These equations are solved exactly for the infinitesimal case. The finite case is then approximated by an incremental superposition of a series of successiye infinitesimal solutions. The results are applied to a plate initially containing a circular hole, and are shown to agree closely with the experimental measurements.     

On the stresses due to an overlapped circular hole on the neutral axis of a deep beam under constant bending moment

Summary In this paper we have discussed an intresting problem on the effect of stress distribution in a deep beam under constant bending moment with a small overlapped circular hole on the neutral axis of the beam. The solution has been obtained in the integral form with the use of bipolar coordinates. Numerical values of the stress have been computed for some points over the edge of the hole.     

Stress concentration factor due to a functionally graded ring around a hole in an isotropic plate

The aim of this work is to present an analytical solution to reduce the stress concentration factor (SCF) around a circular hole in an isotropic homogeneous plate subjected to far-field uniaxial loading. In this paper the elastic response of an inhomogeneous annular ring made of functionally graded material (FGM), inserted around a hole of a homogeneous plate, is studied. By assuming that Young’s modulus varies in the radial direction with power law and that Poisson’s ratio is constant, the governing differential equations for plane stress conditions are obtained. Using stress function a general solution in explicit closed form is presented and the SCF investigated to highlight the inhomogeneity effects. Furthermore, the explicit solution for an inner homogeneous ring, with different properties with respect to those of the plate, is explicitly obtained and numerical results are compared between homogeneous ring and FGM ring.