Limits of Economy of Material in Plates

1955 ◽  
Vol 22 (3) ◽  
pp. 372-374
Author(s):  
H. G. Hopkins ◽  
W. Prager
Keyword(s):  
Yield Condition ◽  
Transverse Load ◽  
Constant Thickness ◽  
Flow Rule ◽  
Plate Material ◽  
Uniform Thickness ◽  
Simply Supported ◽  
Varying Thickness ◽  
Limiting Case ◽  
Associated Flow Rule

Abstract The paper is concerned with the limits of economy of material in a simply supported circular plate under a uniformly distributed transverse load. The plate material is supposed to be plastic-rigid and to obey Tresca’s yield condition and the associated flow rule. The criterion of failure adopted is that used in limit analysis. It is shown that the plate of uniform thickness has a weight efficiency of about 82 per cent. Stepped plates of segmentwise constant thickness are discussed, and the plate of continuously varying thickness is treated as the limiting case obtained by letting the number of steps go to infinity.

A New Method of Analyzing Stresses and Strains in Work-Hardening Plastic Solids

1956 ◽  
Vol 23 (4) ◽  
pp. 493-496
Author(s):  
William Prager
Keyword(s):  
Work Hardening ◽  
Yield Condition ◽  
Transverse Load ◽  
Flow Rule ◽  
Total Stress ◽  
Stress Strain ◽  
Hardening Material ◽  
Characteristic Features ◽  
Associated Flow Rule ◽  
Yield Conditions

Abstract For work-hardening plastic solids, segmentwise linear yield conditions and the associated flow rules constitute a reasonable compromise between the mathematically convenient but physically unsound total stress-strain laws and the physically sound but mathematically inconvenient incremental laws. They allow total stress-strain laws to be used in the small, but retain the characteristic features of incremental laws in the large. The use of a segmentwise linear yield condition and the associated flow rule is illustrated by the analysis of the bending moments and deflections of a simply supported circular plate that is made of a work-hardening material and subjected to a uniformly distributed transverse load.

On the Buckling of Certain Optimum Plate Structures with Linearly Varying Thickness

1959 ◽  
Vol 10 (2) ◽  
pp. 145-148 ◽  
Author(s):  
E. H. Mansfield
Keyword(s):  
Local Buckling ◽  
Constant Thickness ◽  
Sectional Area ◽  
Longitudinal Compression ◽  
Cross Sectional ◽  
Plate Structures ◽  
Buckling Loads ◽  
Simply Supported ◽  
Euler Buckling ◽  
Varying Thickness

SummaryThis paper is concerned with the buckling under uniform longitudinal compression of a variety of structures composed of plates whose thickness tapers linearly to zero across the section. Such structures include the angle of Fig. 1, the strut of cruciform section of Fig. 2 and the simply-supported strip of Fig. 3. For given cross-sectional area and overall dimensions (e.g. length of arm) the sections with linearly varying thickness achieve a greater buckling load (assuming that local buckling, rather than Euler buckling, is the criterion) than sections with any other smooth variation of thickness. These particular sections are therefore optimum sections and, even if they may not be used in practice, provide a convenient yardstick for purposes of comparison. The buckling loads are considerably greater than those for the corresponding “constant thickness” sections.

Optimum Support Position for Flat Plate Subjected to Pressure

1961 ◽  
Vol 65 (612) ◽  
pp. 832-834 ◽  
Author(s):  
R. Kitching
Keyword(s):  
Flat Plate ◽  
Circular Plate ◽  
Normal Pressure ◽  
Constant Thickness ◽  
Bending Stress ◽  
Plate Material ◽  
Concentric Ring ◽  
Simply Supported ◽  
Ring Radius ◽  
Supporting Ring

When a circular plate of constant thickness is simply supported on a concentric ring and is subjected to a uniform normal pressure, there is a radius for the supporting ring giving optimum bending stress conditions in the plate. Assuming the plate deflections are small, it is concluded that the required supporting ring radius varies between 70·1 and 73·0 per cent of the outside radius of the plate, depending on the value of Poisson's Ratio for the plate material.

Finite Elastic-Plastic Expansion of a Cylindrical Tube

1973 ◽  
Vol 2 (4) ◽  
pp. 216-222
Author(s):  
B. Slevinsky ◽  
J. B. Haddow
Keyword(s):  
Plastic Deformation ◽  
Yield Condition ◽  
Cylindrical Tube ◽  
Flow Rule ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
End Conditions ◽  
Cylindrical Tubes ◽  
Limiting Case ◽  
Plastic Flow Rule

A numerical method for the analysis of the isothermal elastic-plastic expansion, by internal pressure, of cylindrical tubes with various end conditions is presented. The Tresca yield condition and associated plastic flow rule are assumed and both non-hardening and work-hardening tubes are considered with account being taken of finite plastic deformation. Tubes which undergo further plastic deformation on unloading are also considered. Expansion of a cylindrical cavity from zero radius in an infinite medium is considered as a limiting case.

Analysis of Creep in Rotating Disks Based on the Tresca Criterion and Associated Flow Rule

1956 ◽  
Vol 23 (2) ◽  
pp. 231-238
Author(s):  
A. M. Wahl
Keyword(s):  
Steady State ◽  
Creep Rate ◽  
Constant Temperature ◽  
Variable Thickness ◽  
Constant Thickness ◽  
Flow Rule ◽  
Test Results ◽  
Rotating Disks ◽  
Tresca Criterion ◽  
Associated Flow Rule

Abstract An analysis of creep deformations in rotating disks based on the Tresca criterion and the associated flow rule is presented. Assuming steady-state creep conditions and a creep rate equal to a function of stress times a function of time, the method is applied to the following cases: (a) Disk with constant thickness and constant temperature, (b) disk with variable thickness and constant temperature, and (c) disk with variable thickness and variable temperature. In many cases, the equations can be expressed in closed form. Comparison is made with test results on rotating disks at elevated temperature as reported in a previous paper. Based on certain stress-creep-rate relations, the method is also applied to the problem of calculating the transient change in stress when the stress distribution changes from an initial to a steady-state condition during the starting period. It is suggested that the simplification effected by the use of these methods may be of value for design purposes pending the development of more accurate methods based on test results.

The Elastic, Plastic Bending of a Simply Supported Plate

1961 ◽  
Vol 28 (3) ◽  
pp. 395-401 ◽  
Author(s):  
G. Eason
Keyword(s):  
Yield Condition ◽  
Constant Load ◽  
Flow Rule ◽  
Plastic Bending ◽  
Simply Supported ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
Perfectly Plastic ◽  
Elastic Perfectly Plastic ◽  
Von Mises

In this paper the problem of the elastic, plastic bending of a circular plate which is simply supported at its edge and carries a constant load over a central circular area is considered. The von Mises yield condition and the associated flow rule are assumed and the material of the plate is assumed to be nonhardening, elastic, perfectly plastic, and compressible. Stress fields are obtained in all cases and a velocity field is presented for the case of point loading. Some numerical results are given comparing the results obtained here with those obtained when the Tresca yield condition is assumed.

On the plastic theory of plates

1957 ◽  
Vol 241 (1225) ◽  
pp. 153-179 ◽  
Keyword(s):  
Plastic Material ◽  
Yield Condition ◽  
Middle Surface ◽  
Transverse Load ◽  
Flow Rule ◽  
Transverse Shear Strain ◽  
Elastic Plates ◽  
Field Equations ◽  
Rigid Plastic ◽  
Tresca Yield Condition

This paper presents a theory of the small deformations of a thin uniform plate under transverse load. The plate is made of non-hardening rigid-plastic material obeying the Tresca yield condition and associated flow rule. The basic assumptions are similar to those made in the conventional engineering theory of thin elastic plates, and the effects of transverse shear strain and rotatory inertia are neglected. Hitherto, the theory has been developed only under conditions of circular symmetry, and the object of the present paper is to remove this restriction. Attention is confined here to the derivation and classification of the field equations. The field equations involve the stress moments and the middle-surface curvature rates as the associated generalized stresses and strain rates. These equations are first referred to Cartesian co-ordinates. The condition of isotropy requires the coincidence of the directions of principal stress moment and curvature rate. One of these two families of directions is characteristic for the equations appropriate to certain plastic régimes. The field equations are therefore referred to curvilinear co-ordinates taken along these directions. A detailed study is made of discontinuities in the field quantities. The field equations are either parabolic or elliptic for the principal plastic régimes.

Some axially symmetric flows of Mohr–Coulomb compressible granular materials

1992 ◽  
Vol 438 (1902) ◽  
pp. 67-93 ◽  
Keyword(s):  
Granular Material ◽  
Granular Materials ◽  
Yield Condition ◽  
Polar Coordinates ◽  
Flow Rule ◽  
Physical Parameters ◽  
Alternative Theory ◽  
Axially Symmetric ◽  
Special Cases ◽  
Associated Flow Rule

In this paper we consider a number of axially symmetric flows of compressible granular materials obeying the Coulomb–Mohr yield condition and the associated flow rule. We pay particular attention to those plastic régimes and flows not included in the seminal work of Cox, Eason & Hopkins (1961). For certain plastic régimes, the velocity equations uncouple from the stress equations and the flow is said to be kinematically determined. We present a number of kinematically determined flows and the development given follows the known solutions applicable to the so-called ‘double-shearing’ model of granular materials which assumes incompressibility and for which the governing equations are almost the same. Similarly, for certain other plastic régimes the stresses may be completely determined without reference to the velocity equations and these are referred to as statically determined flows. In the latter sections of the paper we examine statically determined flows arising from the assumption that the shear stress in either cylindrical or spherical polar coordinates is zero. In the final section we present a numerical solution, which incorporates gravitational effects, for the flow of a granular material in a converging hopper. In addition, we examine the Butterfield & Harkness (1972) modification of the double-shearing model of granular materials which formally includes both the double-shearing theory and the Coulomb–Mohr flow rule theory as special cases. Moreover, for kinematically determined régimes, the velocity equations are the same apart from a different constant, while for statically determined régimes the governing velocity equations are slightly more complicated, involving another constant which is a different combination of the basic physical parameters. Thus some of the solutions presented here can be immediately extended to this alternative theory of granular material behaviour and therefore the prospect arises of devising experiments which might validate or otherwise one theory or the other.

Tresca’s Yield Condition and the Rotating Disk

1983 ◽  
Vol 50 (3) ◽  
pp. 676-678 ◽  
Author(s):  
U. Gamer
Keyword(s):  
Stress Field ◽  
Plastic Strain ◽  
Tensile Stress ◽  
Displacement Field ◽  
Yield Condition ◽  
Rotating Disk ◽  
Flow Rule ◽  
Elastic Plastic ◽  
Solid Disk ◽  
Associated Flow Rule

The displacement field belonging to the elastic-plastic stress field in a rotating solid disk that can be found with the help of Tresca’s yield condition, in textbooks on plasticity, is discontinuous at the elastic-plastic interface. Tresca’s yield condition cannot be applied to this problem since its associated flow rule predicts a negative plastic strain caused by a tensile stress.

A theoretical and experimental study of projectile impact on clamped circular plates

1968 ◽  
Vol 306 (1487) ◽  
pp. 435-447 ◽  
Keyword(s):  
Experimental Study ◽  
Plastic Deformation ◽  
Mild Steel ◽  
Yield Condition ◽  
Flow Rule ◽  
Plastic Strains ◽  
Circular Plates ◽  
Projectile Impact ◽  
Von Mises ◽  
Associated Flow Rule

A method for the analysis of the plastic deformation of a circular plate subject to projectile impact is presented based on the assumption that the material is rigid viscoplastic, obeying a von Mises yield condition and associated flow rule. The predictions of the analysis are com­pared with the results of experiments in which projectiles of different masses are fired at various velocities at clamped plates of mild steel. The plates used in the experiments are such that substantial plastic strains can develop, while the maximum displacements are of the same order as the thickness. The analytical method presented predicts the behaviour of the plates to within the accuracy of the tests. The material constants which fit the results are in accord with those obtained from different tests.

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