Small Deflection Theory Research Articles

Analysis of Thick Circular Plates Undergoing Large Deflections

This investigation considers the effect of transverse shear deformation on bending of the axisymmetrically loaded isotropic and orthotropic circular and annular plates undergoing large deflection. The analysis treats the nonlinear terms of lateral displacement as fictitious loads acting on the plate. The solution of a von Karman-type plate is, therefore, reduced to a plane problem in elasticity and a linear plate-bending problem. Results are presented for simply supported and clamped plates and are in good agreement with the available solutions. For plates considered in this study, the influence of shear deformation on lateral displacement becomes more significant as the orthotropic parameter increases. The linear and nonlinear solutions for orthotropic plates deviate at a low value of the maximum deflection-to-thickness ratio (\Iw\N/\Ih\N). Consequently, the extent of \Iw\N/\Ih\N within which the small-deflection theory is applicable to orthotropic plates is much lower than the value of about 0.4 typically used for isotropic plates, and it depends, in general, on the degree of orthotropy. The technique employed in this study is well suited for the analysis of nonlinear plate problems.

Collapse Behavior and Strength of Steel Silo Transition Junctions. Part II: Parametric Study

This paper presents the results of a major parametric study of the plastic collapse behavior and strength of the complex transition junctions in steel silos using an elastic-plastic large-deflection finite element analysis for axisymmetric shells. Both small-deflection and large-deflection theories are employed to obtain classical limit loads and to clarify the effect of changes in geometry. The collapse loads determined by the finite element analysis are compared with those predicted by a simple hand method derived by Rotter and an improved method developed in a comparison paper. These comparisons show that Rotter’s simple method is satisfactory only for junctions at which the shell segments are of similar thickness. The modified method gives a much closer overall approximation to the finite element results over the wide range of parameters studied. This paper also discusses the alternative possibilities of skirt or hopper collapse when a strong junction is used.

Optimal design of sandwich beams

Minimum weight design of sandwich beams is considered in this paper. A small deflection theory is presented for determining the stresses and deformations in variable thickness elastic sandwich beams that are symmetric about their axes. The face sheets are treated as membranes; the core is assumed to carry only transverse shear. The contribution of the face sheet membrane forces (by virtue of their slopes) to the transverse shear is accounted for in this analysis. A deflection constraint as well as stress constraints are considered. The formulated optimal design problem is solved by the optimization system IDESIGN3 which is based on the Recursive Quadratic Programming Algorithm.

On a one-dimensional theory of finite bending and stretching of elastic plates

The equations of the classical theory of small finite deflections of thin elastic plates are used for the derivation of an approximate one-dimensional theory of plates which are long in comparison with their width. The one-dimensional theory which is obtained is of the eighth-order and includes, in a natural way, the effect of warping stiffness in addition to the effects of stretching, bending and twisting stiffness. The general formulas are specialized so as to obtain a system of equations for the analysis of the lateral buckling of beams which is more general than the previously known results for this class of problems.

Unified Iso-Response Curves for Orthotropic Plates Under Lateral Pressure Pulse

General nondimensional iso-deflection and iso-bending moment curves for the elastic response of orthotropic rectangular plates under lateral pressure pulse are presented. Similarities of loads and plate properties are employed to develop the curves, which represent the peak response for various combinations of orthotropy, density, geometry and loading. The presented curves are valid within the elastic small deflection theory of plates, and their usefulness and limitations are discussed in the paper.

Analysis of natural vibrations of a spherical shell with variable thickness

This paper is concerned with the natural vibrations of a spherical shell with variable thickness in the meridional direction. The governing equations of motion are based upon the classical small deflection theory of thin shells postulated by Love for the case of shell vibrations. In this paper, it is shown that the eight solutions of these equations for the asymmetrical vibrations can be obtained analytically by the method of series solution. As a practical example, four solutions required in the case of a hemispherical shell with a free edge are derived. Last, to demonstrate the utility of the present method, the numerical results are compared with the experimental data.

Plastic collapse of restrained steel silo hoppers

Elevated steel silos and tanks commonly consist of a cylindrical shell, a conical hopper and a skirt. Much of the total weight of the stored material is supported by the hopper, which is in biaxial tension and may fail by forming a plastic collapse mechanism. This paper examines the plastic collapse of hoppers which are sufficiently restrained by a ring at the hopper/cylinder junction for the collapse mode to be entirely confined to the hopper. The hopper joints are assumed to be stronger than the shell plate. An elastic-plastic finite-element program is used to study the plastic collapse behaviour of these hoppers. It is found that the plastic collapse mode is usually a local mechanism near the top of the hopper. Collapse strengths are determined for hoppers of both uniform thickness and varying thickness subject to internal pressure with and without frictional shear. Most of the calculations are performed using elastic-plastic small deflection theory, because this leads to well-defined collapse strengths and relates to classical limit analysis. Calculations made using large deflection theory show that the non-linear changes of geometry have a significant stiffening and strengthening effect. The parametric study presented in this paper defines a simple lower bound to the strength of the hopper.

On Free Vibrations of a Thin Spinning Disk Stiffened With an Outer Reinforcing Ring

Thin spinning annular disks, which have widely varying applications ranging from inertial wheels in spacecraft to computer data storage devices, experience some inherent vibration problems during operation. One of the techniques to control the vibrations of the disk, being analyzed in this paper, is to stiffen it by attaching a reinforcing ring at its outer edge. The present work considers the effect of adding such a ring and discusses the changes in the natural frequencies for a large range of design parameters. The classical plate bending equation based upon small deflection theory which includes the contribution of rotational membrane stresses has been used in the eigenvalue formulation. Numerical results presented in a nondimensional form should be useful in predicting the dynamic response of such a disk stiffened with a circular ring under the spinning conditions.

Bending of Sandwich Plates of Variable Thickness

A small deflection theory, consisting of differential equations and a total potential energy expression, is presented for determining the stresses and deformations in variable thickness elastic sandwich plates symmetric about a middle surface. The theory takes into account the contribution of the face-sheet membrane forces (by virtue of their slopes) to the transverse shear. A finite-difference formulation of the stationary total potential energy principle is presented along with an illustrative application.

Dynamic stability of a rectangular plate subjected to distributed in-plane dynamic force

Dynamic instability of a rectangular plate subjected to an in-plane sinusoidally varying force which varies along the edges is analyzed. The thin plate small deflection theory is used. The boundary conditions of the plate consist of combinations of simply supported edge and clamped edge. Unstable regions are presented for various boundary conditions of the plate and loading conditions.

DYNAMIC STABILITY OF AN INITIALLY DEFLECTED RECTANGULAR PLATE UNDER AN INPLANE DYNAMIC MOMENT

In the following paper, the vibration of an initially deflected rectangular plate under a sinusoidally time-varying inplane moment is examined from the point of view of dynamic instability. The equation of motion describing the large deflection of the initially deflected plate is analyzed by the Galerkin method. The resulting equations for time variables are integrated by using the Runge-Kutta-Gill method. The dynamic instability regions are analyzed by the small deflection theory of a thin plate, neglecting nonlinear terms. The amplitudes of unstable regions are determined by large deflection theory.Numerical results are presented for various shapes and magnitudes of the initial deflection. The effect of the initial deflection on natural frequency, dynamic instability and amplitudes are discussed.

Field Transfer Matrix for Nonlinear Curved Beams

For large‐displacement and small‐strain analysis of planar general arches, the field transfer matrix is formulated for a curved element applicable to arbitrary arch geometry. The total displacements are clearly separated into the large rigid‐body displacements and the small elastic deformations. The formulation includes no approximation except the assumed small deflection theory for the elastic deformations. To establish a general computational strategy, however, the initial arch configuration is assumed to be accurately represented by using blending functions. No special shape functions for displacements are needed. The formulation is shown to be simple and accurate in theory and useful in practice.

Non‐linear boundary value problems for the annular membrane: New results on existence of positive solutions

AbstractThe Föppl‐Hencky theory of small finite deflections has been applied to the study of axisymmetric deformations of annular membranes by various authors. The mathematical problem of uniqueness of tensile solutions for the corresponding non‐linear boundary value problems in the full range of physically meaningful boundary data was solved only recently.3, 4 In this paper, a new integral equation technique of solution is developed which yields the existence of tensile solutions for a parameter range of the boundary data where previous investigations on this matter had not been successful. It is shown that tensile solutions no longer exist, when sufficiently large positive radial displacements are prescribed at the inner edge of the annulus.

Stress analysis of annular sandwich plates of linearly varying thickness

A small deflection theory is presented for stresses and deformations in variable thickness elastic annular sandwich plates that are symmetric about a middle surface. Both the energy expression and the differential equations are developed. In this analysis, the face sheets are treated as membranes, the core is assumed to be inextensible in the thickness direction, to carry only transverse shear stress on its cross sections normal to middle surface, and to be deformable in transverse shear. The theory takes into account the contribution of the face sheet membrane forces (by virtue of their slopes) to the transverse shear.

Effective widths of orthotropic plates loaded uniaxially

An approximate method of evaluating the post-buckling characteristics of orthotropic plates subject to uniaxial compressive loading is presented. The analysis is based on the energy method and small deflection theory and avoids any tedious calculations which would become necessary when the large deflection theory is used. Formulae for effective widths of plates under various edge conditions are derived. Effective width curves for plates with various flexural and torsional properties are presented.

A Structural Model for Molded Thermoplastic Ankle-Foot Orthoses

The structural response of a posterior leaf spring, ankle-foot orthosis (AFO) was studied both experimentally and with a simple theoretical model. The theoretical model, which was compared with other analytical solutions and experimental data, predicted the bending and twisting of the AFO due to unit loads. The simple theoretical model utilized beam equations and small deflection theory. Excellent agreement between test and predicted values was achieved, indicating that the simple theoretical model, which was relatively easy to implement computationally, could serve as the major component of a computer-aided design program.

Application and limitations of the flexural creep test for polymeric materials

Application and limitations of the flexural creep test are described. For this purpose six different thermoplastic materials are tested under low and moderate stress and temperature conditions. The log-log method and a nonlinear creep compliance is used to analyze and compare the data. In order to assess the applicability range of elementary small deflection theory in calculation of strain values, a large deflection analysis is performed. This analysis indicates the presence of an instability condition. Design curves are drawn to reveal the percentage difference between the deflection values obtained on the basis of small and large deflection theories.

The complete process of large elastic-plastic deflection of a cantilever

An extension of the Elastica theory is developed to study the large deflection of an elastic-perfectly plastic horizontal cantilever beam subjected to a vertical concentrated force at its tip. The entire process is divided into four stages: I.elastic in the whole cantilever; II.loading and developing of the plastic region; III.unloading in the plastic region; and IV.reverse loading. Solutions for stages I and II are presented in a closed form. A combination of closed-form solution and numerical integration is presented for stage III. Finally, stage IV is qualitatively studied. Computed results are given and compared with those from small-deflection theory and from the Elastica theory.

Stress Analysis of Sandwich Plates With Unidirectional Thickness Variation

A small deflection theory is presented for the stresses and deformations in variable thickness elastic sandwich plates that are symmetric about a middle surface. In this analysis the face sheets are treated as membranes; the core is assumed to be inextensible in the thickness direction, to carry only transverse shear stress on its cross sections normal to middle surface, and to be deformable in transverse shear. The theory takes into account the contribution of the face sheet membrane forces (by virtue of their slopes) to the transverse shear.

On Large Deflection of Symmetric Composite Plates under Static Loading

The effect of transverse shear deformation on bending of elastic symmetric laminated composite plates undergoing large deformation (in the Von Karman sense) is considered in the present paper. The non-linear terms of the lateral displacement are considered as an additional set of lateral loads acting on the plate. The solution of a Von Karman type plate is therefore reduced to that of an equivalent plate with small displacements. This method offers an alternative technique for obtaining non-linear solutions to plate problems. The solutions of a number of example problems indicate that the non-linear shear deformation theory results, as expected, in higher values of the lateral displacement than the non-linear solutions from the classical plate theory. The difference in the values of the maximum displacement from both solutions, however, remains essentially constant beyond a certain value of the load. It is also noted that the linear and non-linear solutions deviate at a low value of w/h (w = maximum lateral displacement, h = thickness). Consequently, the extent of w/h within which the small deflection theory is applicable to composite plates is much lower than the value of 0.4 typically used for isotropic plates and depends, in general, upon lamination geometry and the degree of anisotropy.