Axisymmetric Bending Research Articles

Stamping and springback of circular plates deformed in hemispherical dies

Circular plates of mild steel with radius-to-thickness ratios between 30:1 and 60:1 were pressed or stamped into a hemispherical die by a matching punch. Measurements of the punch travel, the final curvature of the plates (i.e. after elastic springback) and a record of the wrinkling encountered during pressing are related to the punch force for a range of plate central deflection up to several plate thicknesses. A theoretical analysis, based on axisymmetric bending or stretching of rigid-perfectly plastic plates is presented. This analysis well accounts for the experimental variations in stamping force during pressing and the final curvature of the deformed plates.

Three-Dimensional Elastic Stresses in Thick Plate : Comparison with Classical Plate Theory Solution

In order to estimate the error in the low- and high-order plate theories, three-dimensional elastic stress analysis of a thick plate is carried out. The problem treated here is the axisymmetric bending of a clamped circular plate subjected to an annularly distributed load. For this analysis, use is made of the indirect fictitious-boundary integral method. By the examination of the stress distributions in various thickness ratios, the applicability of the classical plate theory as the lowest-order plate theory is investigated and, further, one suggestion for the applicability of the high-order plate theories is presented.

Fracture propagation resistance of welded nipple joints in axisymmetric repeated static loading

1. In contrast to biaxial loading, the rate of propagation of the crack in repeated loading in the axisymmetric bending of the nipple joint changes in accordance with a curve with a maximum; this is explained by the effect of the stress gradient in thickness. 2. An increase in the diameter of the surface cylindrical crack displaces the maximum of the KI=f(a) curve to longer crack lengths. 3. In evaluating the endurance of pressure vessels, the duration of the stage of failure propagation should be carried out for biaxial tensile loading, which gives a conservative evaluation of the “life” of the structures.

Axisymmetric Flexure of Circular Sandwich Plate Including Transverse Shear in Facings

AbstractAn approximate formulation is presented for the bending of an isotropic sandwich circular plate under transverse load taking into account shear forces in the core and facings. The equations are solved in terms of Bessel function and results are computed for axisymmetric bending, when the outer edge is clamped.

Nonlinear bending of circular sandwich plates

In this paper, fundamental equations and boundary conditions of nonlinear axisymmetrical bending theory for the circular sandwich plates with a soft core are derived by means of the method of calculus of variations. Especially in the case of very thin faces, the preceding fundamental epuations and boundary conditions simplity considerably. For example, a circular sandwich plate with edge clamped but free to siip under the action of uniform lateral load is considered. A more accurate solution of this problem has been obtained by means of the modified iteration method.

Nonlinear axisymmetric bending of annular plates with varying thickness

Finite-element analysis of the large deflection bending of annular plates with variable thickness is presented. The more general Reissner plate equations as well as von Karman plate equations are used in the formulation. Stresses and deformation results are presented showing the effects of radius-to-thickness ratio (i.e. shear deformation), nonlinearity and material orthotropy. The finite-element solutions are found to be in good agreement with available exact and approximate analytical solutions.

Technical note: On the equivalence of axisymmetric bending, thermal, and autofrettage residual stress fields

Technical note: On the equivalence of axisymmetric bending, thermal, and autofrettage residual stress fields

Finite Element Analysis of Mindlin Plates

A finite element displacement formulation with the shear deformation capability for the axisymmetric bending analysis of circular plates is presented. The displacement field is expressed in terms of interpolation functions which are exact solutions to Mindlin’s governing plate equations. Numerical results for some representative example problems are also presented to demonstrate the accuracy of the proposed method.

Dynamic stability of circular cylindrical shells under periodic compressive forces

Based on the Donnell equations modified with the transverse inertia force, the dynamic stability of circular cylindrical shells under both static and periodic compressive forces is theoretically analyzed under four different boundary conditions, with the effect of the axisymmetric unperturbed bending vibration taken into consideration. The problem is first reduced to that of a finite degree-of-freedom system with the Galerkin procedure, the stability of which is examined by using Hsu's method. Calculations are carried out for typical cases and the instability regions of the principal, secondary and combination parametric resonances are determined for the frequency range covering up to several times the lowest natural frequency. It is found, among others, that the effect of the unperturbed motion is quite significant for shells with moderate length while that of the longitudinal resonance is generally negligible for thin shells.

Bending of thick plates under an arbitrary load: PMM vol. 41, n≗ 5, 1977, pp. 909–914

A problem of bending of a plate under an arbitrary load is studied. A method proposed by Lur'e [1] is employed, which reduces the three-dimensional problems of the plate theory to the two-dimensional problems using the infinite order differential operators of the displacements and rotations of the middle plane of the plate. Three stress functions are introduced, and these make it possible to consider the effects of the normal and tangential loads separately. Approximate equations are constructed using the method of homogeneous solutions [2]. a problem of axisymmetric bending of a plate under the action of tangential forces is solved using the first approximation equations.

Axisymmetric bending vibrations of a mildly sloping helicoidal shell

Axisymmetric bending vibrations of a mildly sloping helicoidal shell

Finite element analysis of laminated composite shell junctions

The axisymmetric bending of laminated composite shell junctions have been analysed by the finite element displacement method using laminated anisotropic shell theory. Numerical results have been presented for cylinder-cone shell junction and cylinder-geodesic-isotensoid dome junction. These results have been compared with results for isotropic case.

Nonlinear Bending of Circular Plates

The existence of a solution to the Von Karman equations for the nonlinear axi-symmetric bending of a circular plate is proved. The method is related to that used by Hammerstein in treating nonlinear integral equations.

Buckling of orthotropic torispherical pressure vessels

Asymmetric buckling analysis of orthotropic pressure vessels, subjected to uniform external pressure is carried out. Sanders' nonlinear thin shell theory is employed to derive the buckling equations. The prebuckling state is represented by a linear axisymmetric bending state. The differential equations governing the prebuckling and buckling states are solved by numerical integration method using fourth order Runge-Kutta scheme. Results are presented for cylindrical pressure vessels with: (1) spherical attachment (2) torispherical head. The effect of L/R ratio ( L and R are length and radius of cylinder) on the buckling behaviour is investigated.

Equations of the axisymmetric bending of three-layered shells of revolution

Equations of the axisymmetric bending of three-layered shells of revolution

Comparison of elasticity theory and shell theory solutions

The axisymmetric bending of cylinder-cone shell junction has been analysed by the finite element method using three dimensional elasticity theory. Numerical results have been worked out for three shells having cylinder diameter to thickness ratios of 41, 21 and 13. These results have been compared with the results obtained by the classical thin shell theory of Sanders. The validity of the assumptions made in the classical thin shell theory is discussed. The nature of singularity at the re-entrant corner has been studied.

A collocation solution of the nonlinear equations for axisymmetric bending of shallow spherical shells

A collocated load is developed to replace a theoretically prescribed load to obtain an approximate solution to the nonlinear shallow shell problem. Any axisymmetric load, continuous or noncontinuous, may be treated together with any symmetrical edge condition-clamped, simple, roller or elastic support. Convergence is not proved and a solution is judged acceptable when the residuals at noncollocated points meet arbitrarily imposed smallness criteria. By meeting these criteria an equipollent load is obtained with the associated deformations satisfying the conditions of compatibility to within a small prescribed error. Examples of load-deflection curves, stress and deflection curves and residual errors are given.

Transient axisymmetric bending stress in an infinite cylindrical shell at an elastic ring stiffener

Transient axisymmetric bending stress in an infinite cylindrical shell at an elastic ring stiffener

Elastic bending of circular plates of variable thickness: An analytical and experimental study

The stress distribution in thin annular and circular plates, in which the thickness varies as any piecewise continuous function of the radius r, is investigated. The plate may be either stepped or continuous, having simply supported, free or rigidly clamped edge constraints. The types of loading considered are those which result in either axisymmetric or antisymmetric bending (bending about a diameter) of the plate. A simple finite-element approach which considers the plate to be composed of concentric rings of uniform thickness is used, giving results which are in very good agreement with known exact solutions for plates of varying thickness. Experimentally determined stress distributions in several models of varying thickness substantiate the applicability of the analytical method and indicate the limitations encountered in using this theory.

Numerical Method for Nonlinear Axisymmetric Bending of Arbitrary Shells of Revolution and Large Deflection Analyses of Corrugated Diaphragm and Bellows

A numerical method which can be applicable to the nonlinear axisymmetric bending of arbitrary shells of revolution is proposed. It is based on the iterative scheme and the finite difference approximation, and is found to show good convergence and to be practical. Using this method, the large deflection problems of the corrugated diaphragm and bellows are solved. The results show the unnegligible nonlinear effects and the limitations of the linear theory and the approximate nonlinear solution now used.