Buckling of a Thin Annular Plate Under Uniform Compression

1958 ◽  
Vol 25 (2) ◽  
pp. 267-273
Author(s):  
N. Yamaki
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Critical Loads ◽  
Equilibrium Equation ◽  
Annular Plate ◽  
Elastic Stability ◽  
Infinite Strip ◽  
Uniform Compression ◽  
The Stability

Abstract This paper deals with the elastic stability of a circular annular plate under uniform compressive forces applied at its edges. By integrating the equilibrium equation of the buckled plate, the problem is solved in its most general form for twelve different combinations of the boundary conditions of the edges. For each case cited the lowest critical loads are calculated with the ratio of its radii as the parameter. It is clarified that the assumption of symmetrical buckling, which has been made by several researchers, often leads to the overestimate for the stability of the plate. Discussions for the limiting cases of the circular plate and infinite strip also are included.

scholarly journals The elastic stability of an annular plate

1924 ◽  
Vol 106 (737) ◽  
pp. 268-284 ◽  
Keyword(s):  
Annular Plate ◽  
Practical Interest ◽  
Middle Surface ◽  
Elastic Stability ◽  
Thin Plates ◽  
Thin Shells ◽  
Stability Equation ◽  
Inner Radius ◽  
Plane Plate ◽  
The Stability

1. Introduction and Summary. —This paper deals with the elastic stability of a circular annular plate under uniform shearing forces applied at its edges. Investigations of the stability of plane plates are altogether simpler than those necessary in the case of curved plates or shells. In the first place, as shown by Mr. R. V. Southwell, two of the three equations of stability relate to a mode of instability that is not of practical interest, and are entirely independent of the third equation which gives the ordinary mode of instability resulting in the familiar bending of the middle surface of the plate. Consequently with a plane plate there is only one equation of stability to be solved, as contrasted with the case of a shell where the three equations are dependent, and must all be solved. In the second place the theory of thin shells can be used with confidence in a plane plate problem, though a more laborious procedure is necessary to deal adequately with a shell. The only stability equation required for the annular plate is therefore deduced without trouble from the theory of thin shells, and its solution presents no difficulty in the case of uniform shearing forces. A numerical discussion is given of the stability of the plate under such forces, the “favourite type of distortion” and the stess that will produce it being obtained for plates with clamped edges in wich the ratio of the outer to the inner radius exceeds 3·2. To some extent to results have been checked by experiment, in which part of the work the viter is indebted to Prof. G. I. Taylor for his valuable help and advice. Distrtion of the type predicted by the theory took place in the two thin plates of rober different ratio of radii, which were used. The disposition of the loci of points which undergo maximum normal displace nt gives some idea of the appearance of the plate after distortion has taken pce. The points have been calculated for a plate in which the ratio of radii 4·18, and the loci are shown on a diagram, which may be compared with a potograph of a distorted plate in which this ratio is 4·3. The ratio of normal dplacements of points of the plate can be seen from contours drawn on the ne diagram. (See pp. 280, 281.)

Stability of laminated composite and sandwich FGM shells using a novel isogeometric finite strip method

2019 ◽  
Vol 37 (4) ◽  
pp. 1369-1395 ◽  
Author(s):  
Mohammad Amin Shahmohammadi ◽  
Mojtaba Azhari ◽  
Mohammad Mehdi Saadatpour ◽  
Saeid Sarrami-Foroushani
Keyword(s):  
Boundary Conditions ◽  
Critical Loads ◽  
Laminated Composite ◽  
Functionally Graded ◽  
Laminated Shells ◽  
Content Type ◽  
Finite Strip ◽  
Strip Method ◽  
Finite Strip Method ◽  
The Stability

Purpose This paper aims to analyze the stability of laminated shells subjected to axial loads or external pressure with considering various geometries and boundary conditions. The main aim of the present study is developing an efficient combined method which uses the advantages of different methods, such as finite element method (FEM) and isogeometric analysis (IGA), to achieve multipurpose targets. Two types of material including laminated composite and sandwich functionally graded material are considered. Design/methodology/approach A novel type of finite strip method called isogeometric B3-spline finite strip method (IG-SFSM) is used to solve the eigenvalue buckling problem. IG-SFSM uses B3-spline basis functions to interpolate the buckling displacements and mapping operations in the longitudinal direction of the strips, whereas the Lagrangian functions are used in transverse direction. The current presented IG-SFSM is formulated based on the degenerated shell method. Findings The buckling behavior of laminated shells is discussed by solving several examples corresponding to shells with various geometries, boundary conditions and material properties. The effects of mechanical and geometrical properties on critical loads of shells are investigated using the related results obtained by IG-SFSM. Originality/value This paper shows that the proposed IG-SFSM leads to the critical loads with an approved accuracy comparing with the same examples extracted from the literature. Moreover, it leads to a high level of convergence rate and low cost of solving the stability problems in comparison to the FEM.

scholarly journals V. On the general theory of elastic stability

1914 ◽  
Vol 213 (497-508) ◽  
pp. 187-244 ◽  
Keyword(s):  
General Theory ◽  
Boundary Conditions ◽  
Radial Pressure ◽  
Elastic Stability ◽  
Circular Ring ◽  
Theory Of Elasticity ◽  
Comprehensive Theory ◽  
The Subject ◽  
The Stability ◽  
Straight Rod

Problems which deal with the stability of bodies in equilibrium under stress are so distinct from the ordinary applications of the theory of elasticity that it is legitimate to regard them as forming a special branch of the subject. In every other case we are concerned with the integration of certain differential equations, fundamentally the same for all problems, and the satisfaction of certain boundary conditions; and by a theorem due to Kiechiioff we are entitled to assume that any solution which we may discover is unique. In these problems we are confronted with the possibility of two or more configurations of equilibrium , and we have to determine the conditions which must be satisfied in order that the equilibrium of any given configuration may be stable. The development of both branches has proceeded upon similar lines. That is to say, the earliest discussions were concerned with the solution of isolated examples rather than with the formulation of general ideas. In the case of elastic stability, a comprehensive theory was not propounded until the problem of the straight strut had been investigated by Euler, that of the circular ring under radial pressure by M. Lévy and G. H. Halphen, and A. G. Greenhill had discussed the stability of a straight rod in equilibrium under its own weight, under twisting couples, and when rotating.

scholarly journals The elastic stability of a thin twisted strip—II

1937 ◽  
Vol 161 (905) ◽  
pp. 197-220 ◽  
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Main Part ◽  
Fourier Expansion ◽  
Elastic Stability ◽  
Numerical Work ◽  
The Stability ◽  
Good Agreement

I—In a previous paper the present writer discussed both theoretically and experimentally the equilibrium and elastic stability of a thin twisted strip, and the results obtained by the theory were found to be in good agreement with observation. It has, however, been pointed out by Professor Southwell, F. R. S., that the solution of the stability equations which was given in that paper may only be regarded as an approximate solution for, although it satisfies exactly the differential equations and two boundary conditions along the edge of the strip, it only satisfies the two remaining boundary conditions approximately. The author has also noticed that the coefficients n a m in the Fourier expansion of θ 2 cos mθ which were used in A are incorrect when m = 0, and this has led to errors in the numerical work so that the values of ᴛb 2 / π 2 h which are given in Table I of A are wrong. In the present paper a solution of the stability equations is obtained which satisfies all the boundary conditions. This solution is very much more complicated than the approximate solution and much greater labour is required for the numerical work. The numerical work for the approximate solution of A has also been revised and the corrected results are given in 9, 10. It is found that the results for the approximate solution are in good agreement with those obtained from the exact solution and that both agree moderately well with the experimental results which are given in A. The main part of this paper is an extension of the previous work and is concerned with the stability of a thin twisted strip when it is subjected to a tension along its length. The theory has been compared with experiment and satisfactorily good agreement between them was found.

Stability Analysis of the Stiffened Plate with Rids under the Longitudinal Loads

2011 ◽  
Vol 243-249 ◽  
pp. 279-283
Author(s):  
Yu Zhang
Keyword(s):  
Potential Energy ◽  
Boundary Conditions ◽  
Critical Loads ◽  
Minimum Condition ◽  
Stiffened Plates ◽  
Stiffened Plate ◽  
Simply Supported ◽  
Total Potential ◽  
The Stability ◽  
Work Done

The stiffened plate with rids was considered as a whole structure. Using energy method the stability of stiffened plates with rids under the longitudinal forces was analyzed. Calculating the potential energy of deformation of plate and that of rids and the work done by the neutral plane forces of plate when the plates were buckled, the formulas of critical loads of the stiffened plate with rids under longitudinal forces were derived from the minimum condition of total potential energy. Using the formulas in this paper engineers can easily calculate the critical loads of the stiffened plate with rids under the boundary conditions: the opposite sides are fixed and the other opposite sides are simply supported, four sides are simply supported. The formula of critical loads of the stiffened plate with rids under other boundary conditions can be derived using the method in this paper.

The Behaviour of a Clamped Circular Plate in Compression

1961 ◽  
Vol 12 (1) ◽  
pp. 51-64 ◽  
Author(s):  
A. N. Sherbourne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Circular Plate ◽  
Digital Computer ◽  
Large Deflection ◽  
Theoretical Solution ◽  
Uniform Compression ◽  
Simply Supported ◽  
Von Karman

SummaryA theoretical solution is presented for the problem of the clamped circular plate loaded in uniform compression. The solution employs a numerical method programmed for a digital computer. Instead of solving the classical von Kármán large deflection equations, a step-by-step integration of the elastic differential equations of equilibrium is carried out until suitable boundary conditions are attained. The method is an extension of one developed earlier to explain the behaviour of the simply-supported plate.

On the Elastic Stability of a Rectangular Plate, when Subjected to a Variable Edge Thrust

1935 ◽  
Vol 31 (3) ◽  
pp. 368-381 ◽  
Author(s):  
D. M. A. Leggett
Keyword(s):  
Boundary Conditions ◽  
Approximate Method ◽  
Rectangular Plate ◽  
General Problem ◽  
Elastic Stability ◽  
Detailed Consideration ◽  
Various Boundary Conditions ◽  
The Stability

The stability of a rectangular plate, subjected to constant thrust over opposite pairs of edges, has been treated with some degree of completeness for various boundary conditions. The more general problem, in which the thrusts are no longer constant, has not yet received any treatment apart from the approximate method developed by E. Schwerin†, which would appear to be capable of only limited extension. The object of this paper is accordingly the detailed consideration of a simple case when the thrust is no longer constant.

scholarly journals Stability of orthotropic plates

2020 ◽  
Vol 166 ◽  
pp. 06004
Author(s):  
Mykola Surianinov ◽  
Dina Lazarieva ◽  
Iryna Kurhan
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Boundary Conditions ◽  
Critical Loads ◽  
Orthotropic Plate ◽  
Algebraic Equations ◽  
Stability Equation ◽  
Element Analysis ◽  
Linear Algebraic Equations ◽  
The Stability

The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.

scholarly journals The elastic stability of a long and slightly bent rectangular plate under uniform shear

1937 ◽  
Vol 162 (908) ◽  
pp. 62-83 ◽  
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Elastic Stability ◽  
Stability Problems ◽  
Uniform Shear ◽  
Lines Of Curvature ◽  
The Stability

1—The problem of the elastic stability of a plane rectangular plate when subjected to uniform shear has been approximately solved for various conditions (Cox 1933; Timoshenko 1936). In the case of an indefinitely long strip an exact solution has been found (Southwell and Skan 1924), but it appears that no attempt has been made to investigate what happens if the plate is no longer plane. It is accordingly the object of this paper to consider the stability of a long strip, slightly curved, when its two side edges are subjected to uniform shear. 2—In what follows we assume that the thickness and curvature of the plate are constant, and that the edges of the plate are two generators and two lines of curvature. It is, moreover, further assumed that the plate is thin as in all similar stability problems, and that it is of such length that the boundary conditions over the two curved ends can be ignored.

Buckling of Circular Cylindrical Shells Using a Displacement Function

1974 ◽  
Vol 96 (4) ◽  
pp. 1322-1327
Author(s):  
Shun Cheng ◽  
C. K. Chang
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Displacement Function ◽  
Combined Loading ◽  
Trial Solution ◽  
Governing Differential Equation ◽  
Circular Cylindrical Shells ◽  
The Stability

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

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