Pure Bending Problem Research Articles

Equations of finite bending of thin-walled curvilinear tubes

A system of nonlinear equations proposed by E. Reissner for the problem of elastic pure bending of thin‐walled curvilinear tubes is considered. The formulation of the problem is refined, and the numerical solution of the equations obtained is compared with the finite‐element results.

A mathematical analysis of the elasto-plastic plane stress problem of a power-law material

In this paper a generic study on the plane stress problem of a power-law material undergoing infinitesimal deformations is carried out, and a general solution for the stress and strain fields is derived using a stress function method and analytic function theory. Hencky's deformation theory and von Mises' yield criterion are used, and a differential transformation is invoked in the analysis. As an example, the closed-form solution of the pure bending problem of a thin beam of power-law material is obtained by applying the general solution directly.

Axisymmetric Elastic-Plastic FE Analysis of Pressurized Elbows

Abstract A variational formulation for the pure in-plane bending problem of pressurized elbows with arbitrary cross section, written in the 3-D infinitesimal displacement continuum mechanics framework, and particularly appropriate when a refined elastic-plastic local response is needed (e.g., in the presence of ratcheting), is presented. The similarity between this problem and an axisymmetric analysis is underlined and turned to account to solve the former using existing FE routines devoted to the latter. Some applications are given to illustrate the potentialities of the model.

Finite deformation elasto-plastic solution for the pure bending problem of a wide plate of elastic linear-hardening material

An exact elasto-plastic analytical solution for a finitely deformed plane strain wide plate of elastic linear-hardening material subjected to pure bending is derived in this paper using a tensorial formulation. This solution is based on a finite-strain version of Hencky's deformation theory, the von Mises yield criterion and the incompressibility assumption. The Hencky (logarithmic) strain tensor is adopted to measure the finite deformations, with the undeformed flat plate as the reference configuration. The constitutive relations are derived in tensor form using the Hencky strain and the Cauchy stress. With both the linear-elastic and strain-hardening-plastic material responses included, the present solution can represent the whole bending process of the plate from its initial flat state to its final curved state with arbitrarily large deformation. It is shown that this solution exhibits general characteristics, from which three specific solutions of practical value can be obtained. Being expressed in both tensor and component forms, the present exact solution for the stress and strain fields furnishes a new and systematic analytical pattern for the elasto-plastic analysis and strength design of a strain-hardening wide plate subjected to large-deformation pure bending.

Pure bending problem for a vibrating elastic plate with non-uniform temperature

Pure bending problem for a vibrating elastic plate with non-uniform temperature

Stress intensity factors for cracks emanating from a circular hole in a circular disk under diametral compression.

Stress intensity factors for cracks emanating from a circular hole in a circular disk under diametral compression are analyzed by the body force method. The stress intensity factors K1 for several cases of the ratio β of the inner to the outer radius and the ratio λ of crack length to the inner radius are numerically calculated. The main results are summarized as follows : (1) In the case of λ≤0.1, an approximate equation for K1 on the basis of the maximum tensile stress at the edge of a hole is derived. (2) In the case of β≤0.3 and λβ≥0.1, K1 can be estimated by the results of a circular disk containing a central crack without a hole. (3) With increasing β, K1 approaches the results for the pure bending problem where the width is equal to the difference between the inner and the outer radius.

A note on the problem of pure bending for linear elastic materials with voids

This note concerns the problem of quasi-static pure bending of a beam in the context of the complete theory of linear elastic materials with voids presented in [1]. It is shown here that the solution in the context of the complete theory of [1] is coincident with the pure bending solution of classical elasticity for small time, and that the solution for large time is the bending solution given in [1], a solution which neglected the rate effect in the complete theory of [1]. In between these two limit solutions the rate effect moderates a monotonic transition.

On the bending of a plane inhomogenbous curved beam: PMM vol. 43, no. 1, 1979, pp. 182–183

The plane problem of pure bending of a uniformly inhomogeneous curved beam bounded by two arcs of concentric circles and two radii, is considered. It is assumed that the material of the beam is isotropic, has a constant Poisson's ratio and a coordinate-dependent Young's modulus. It is shown that when the Young's modulus is defined in a specified manner, men the problem has an exact, elementary solution, and the solution is given.

On stretching, twisting, pure bending and flexure of pretwisted elastic plates

Previously derived results for the mentioned canonical load cases for pretwisted rectangular plates (or shallow helicoidal shells) of rectangular cross section are generalized so as to include arbitrarily prescribed cross wise thickness variations. The present method of derivation is simpler than the earlier one in terms of Marguerre's equations for transverse deflection and Airy stress function, and consists in applying a semi-inverse solution procedure to the system of equilibrium and compatibility equations of shallow shell theory. A particular case of the results of this paper concerns the problem of pure bending of a pretwisted plate with elliptical thickness variation. The results for this case coincide with recent results obtained by Goodier and Griffin through use of three-dimensional elasticity theory.

The Side-Force Problem for Shallow Helicoidal Shells

The side-force problem for a shallow helicoidal shell is shown to be the complete static-geometric analog of the pure bending problem for the same shell. The solution of the former in terms of elementary functions is obtained, without another set of independent calculations, simply by applying the rules of the static-geometric duality to that of the latter. The analogy also enables us to use the same computer program developed for the pure bending problem (without any modification) to generate numerical results for the side-force problem.

A note on pure bending of a piezoelectric rectangular plate

The problem of pure bending of a piezoelectric rectangular plate is considered in this paper by using the electromagnetic equations of Maxwell and equations of elasticity along with the constitutive equations of a piezoelectric quartz.

Pure Bending of Shallow Helicoidal Shells

The solution of the problem of pure bending of shallow helicoidal shells is obtained in terms of elementary functions. Numerical results for the direct and bending stresses as well as for the overall stiffness coefficient of the shell are given for various pitch-to-thickness ratios. An interesting aspect of the analysis is that the displacement state of the shell is multivalued in the polar angle θ while the stress state is unsymmetric but periodic in θ. Moreover, unlike the problem of axial torsion and extension where there is no edge effect, the stress state for the present problem exhibits both an interior state and an edge zone state.

On pure bending of a curved bar of piezoelectric material

The present note makes use of the electromagnetic equations of Maxwell, equations of elasticity coupled with the constitutive relations of a piezoelectric quartz to solve the piezoelectric analogue of the elastic problem of pure bending of a curved bar.

On the Plastic Bending of a Thin Strip in its Plane

The problem of pure bending of a thin strip in its plane is treated as one of generalized plane stress for a strainhardening rigid plastic material. The method is based upon the deformation theory of plasticity, and the material is assumed to follow the yield condition of von Mises. The solution can be obtained solving a basic non-linear differential equation numerically. Especially for the rigid perfectly plastic material, a heuristic solution is found in closed form. Some numerical results are presented and an approximate expressions for stress and strain distributions are derived. These results are compared with experimental measurements.Consequently, it is shown that the effect of any strainhardening characteristic on the strain distribution is very small, but is large on the stress distribution and the foregoing analysis is useful for practical purpose.

On Finite Bending of Pressurized Tubes

Abstract A unified treatment is presented of two well-known problems which have until now been considered separately. The two problems are: (a) the linear problem of pure bending of curved tubes, and (b) the nonlinear problem of pure bending of straight tubes. In both problems the effect of uniform internal pressure is included. The essential step in the present analysis is the treatment of the flattening of the cross sections of the tube by means of a theory of finite bending of circular rings. The general results of the paper are used to obtain improved values for the stability parameters in the problem of flattening instability of originally straight tubes acted upon by end bending moments, and also to obtain results on the effect of slight original curvature of the beam axis in the problem of flattening instability.

Pure bending of pretwisted rectangular plates

An explicit solution is obtained to the problem of pure elastic bending of a pretwisted rectangular plate. The results are generalizations of known simple solutions for plane stress and for transverse bending of flat plates. They may have application to turbine and compressor blades and to aircraft propellers.

Bending With Axial Force of Curved Bars in Plasticity

Abstract The problem of symmetrical pure bending with axial force of a curved bar in plasticity is considered. A method is given for finding the axial load and bending moment which produce a given strain distribution. This method is based upon approximating the stress-strain curves by means of broken lines. By increasing the number of sides of these broken lines it is possible to solve our problem with as high accuracy as is desired.