Analysis of cross and vertical buckling in sheet metal rolling

In sheet metal rolling, shape defects called “cross buckling” or “vertical buckling” sometimes appear, which are wrinkles like washboards. The direction of the crest line of the cross buckling inclines at a certain degree against the rolling direction, while that of the vertical buckling is parallel to the rolling direction. In this study, analysis of the cross and vertical buckling is performed using the elementary theory of buckling. First, we calculate the stress distribution in three-dimensional sheet metal rolling near the exit cross section inside the roll gap. Next, we calculate the residual stress distribution near the exit cross section outside the roll gap. Furthermore, sheet metal buckling is analysed using the residual stress distribution. Type of buckling, distance between neighboring wrinkles, inclination of the crest line of the wrinkles against the width direction and the region where the wrinkles appear are obtained. We compare analytical results with published experimental results, and find that the former agree well with the latter. Hence, we conclude that this method of analysis is valid, and that the cause of the cross and vertical buckling is the residual stress distribution near the exit cross section outside the roll gap.     

Classical buckling of a thin-walled tube subjected to bending moment and internal pressure

The classical buckling of a thin-walled circular cylindrical shell which is subjected simultaneously to uniform bending moment and internal pressure is investigated in an approximate fashion by assuming an appropriate eigenmode and obtaining a “best-fit” solution of the resulting equations. In this way, explicit formulae are derived for critical loading combinations and for the characteristics of the modal form. These formulae agree reasonably well with a wide range of numerical data previously obtained by Seide and Weingarten. A feature of the scheme is that it is feasible to interpret the results in physical terms, and to envisage the roles of bending, twisting and stretching in the critical solution.     

A study of the plastic buckling of axially compressed cylindrical shells with a thick-shell theory

A thick shell theory is used to calculate the critical load of plastic buckling of axially compressed cylindrical shells. The buckling equations are derived with the principle of virtual work on the basis of a transverse shear deformable displacement field. The deformation theory of plasticity is used for constitutive equations. To fit the uniaxial stress–strain curve, the Ramberg–Osgood equation is used. In the numerical examples special attention is paid to the dependence of the buckling mode on the ratios of radius to thickness R/h and length to radius L/R. This dependence divides the (R/h,L/R)-plane into simply connected regions each of which corresponds to a buckling mode. These regions form a “buckling mode map”.     

Specific Features of the Behavior of the Coercive Force in Low-Carbon Plastically Deformed Steels

The relationship between the coercive force in low-carbon steels under plastic extension and compression and the values of deformation and actual and residual stresses are studied. This relationship is investigated for both “ slow” loading (when an equilibrium deformation is attained for each load value) and “fast” loading (when such equilibrium is not attained). It is shown that (i) a comparatively small increase in the coercive force in a loaded condition is due only to an increase in the density of dislocations in the process of plastic extension; (ii) a significant steep increase in the coercive force accompanying removal of the load from a plastically stretched specimen is fully due to residual compression stresses; (iii) the values of the coercive force under “slow” and “fast” loading are significantly different in the region of small deformations less than 2.5%; (iv) these values are close to each other in the loaded state for all deformations up to 10%; (v) a relief of the compression stress that creates plastic deformations causes a steep decrease in the coercive force that is as large as its increase following relief of plastic extension; this is explained by the emergence of a significant residual tension stress. The obtained results are of importance for the use of the method based on measuring the coercive force to test steel structures under the conditions when plastic deformations develop.__________Translated from Defektoskopiya, Vol. 41, No. 5, 2005, pp. 24–38.Original Russian Text Copyright © 2005 by Kuleev, Tsar’kova, Nichipuruk.     

Collision between a ring and a beam

With car–parapet collision accidents in mind, a normal collision between a free-flying half ring and a simply supported beam with/without axial constraints is studied, in which an elastic–plastic half ring with an attached mass and the elastic–plastic beam are taken as the simplest models of a car and a parapet, respectively. Particular attention is paid to the energy partitioning between the two structures and the evolution of the contact regions during collision. A mass–spring finite difference (MS–FD) model is employed whilst the large deflection and axial stretching/compression are incorporated. The numerical results show that the less stiff (i.e. softer) structure will dissipate more energy and the contact regions will move away from the initial contact points. With the increase of the relative thickness of the beam to the ring, the final deformation of the half ring will transform from a “U” shape to a “W” shape.     

New results for the fretting-induced stress concentration on Hertzian and flat rounded contacts

Recent work on fretting fatigue has emphasized the role of stress concentration on fretting damage, while previous work had concentrated on empirical parameters to assess influence of fretting on fatigue life. In particular, analogies with fatigue in the presence of a crack or a notch have been noticed, suggesting that the stress field induced by frictional contact per se may explain the reduction of fatigue life due to fretting.In the paper, new analytical and numerical solutions are produced for the stress concentration induced in typical fretting contacts involving the Hertzian geometry or the flat punch with rounded corners in view of application to the dovetail joints. Normal and tangential load (in the Cattaneo–Mindlin sense) is considered with “moderate” or “large” bulk stresses.     

Post-collapse characteristics of ductile circular honeycombs under in-plane compression

The post-collapse behaviour of a circular honeycomb material under in-plane compression is analysed in order to estimate the effect of the structural topology on the material strength. A structural approach using the limit analysis and the concept of an equivalent structure is employed to describe the large plastic deformations during post-collapse process. Based on some published experimental results (International Journal of Solids Structures 36 (1999) 4367–996) and our numerical simulations, certain deformation patterns are constructed depending on the direction of loading, and the corresponding post-collapse load-carrying capacities during large deformation until densification of cells are presented.The present analysis shows that the post-collapse stress associated with an equi-biaxial compression is not excessively larger than the corresponding uniaxial stresses, in contrast to those of hexagonal honeycombs in response to biaxial loading. This behaviour is attributed to the different deformation mechanism as the curvature of a cell wall invokes bending without stretching.The influence of the size of the connecting segment between two neighbouring cells is studied, showing that the “shape” of the limit surface varies significantly depending on this connection.     

The objective function construction for the multi-criteria problems of cyclic bending

A new method for the single objective function construction for multi-criteria optimization problems is presented. This objective function estimates the “distance” between the required and actual distributions of the inelastic (thermal and plastic) strains in the body. In some specific cases such as the decreasing of the mechanical imperfection of long-dimensional profiles, the objective function may contain the initial curvature or the initial stress distribution as parameters. Several applied problems such as wire levelling and beam straightening are solved and discussed.     

Geometry of “developable cones”

When a thin disc is supported on the rim of a bowl, and its centre is pushed down by a finger, it adopts a characteristic conformation, known as a “developable cone”, and sketched in Fig. 1(a): the main, broadly conical, shape can only form if about one-quarter of the disc buckles upwards. There is a curved intersection between the two parts, which takes the form of a crescent-shaped “crease” near its apex, but with the flanking regions less tightly deformed. The “developable cone” is a recurring motif in a wide range of physical situations—crumpling, buckling, draping—and its mechanics provides a key to understand the phenomena, whether the disc deforms in the elastic or the plastic range. The task of this paper is to study only geometrical features of the “developable cone”. The first step is to replace the actual crease (Fig. 1(a)) by an idealised “sharp” crease (Fig. 1(b)). The second step is to study the apparently “large-rotation” problem of kinematics by means of an adaptation of the classical “yield-line” pattern of folding, but with a crucial added constraint that springs from Gauss's analysis of inextensional deformation. We illustrate the method via a graded sequence of examples, and we close with a discussion.     

The secondary buckling and post-secondary-buckling behaviours of rectangular plates

Secondary buckling and post-secondary-buckling behaviours are theoretically studied for simply-supported rectangular plates, whose primary buckling modes of deflection contain more than half-waves in the load acting direction. Modal coupling effects more complex than one two-term-coupling effects are incorporated into the secondary buckling and post-secondary-buckling analyses. Then, unstable or stable symmetric secondary branching points are found on the post-primary-buckling paths and “snap through” motions involving an abrupt change in wave-form are shown to be possible. Wave-form variations along post-secondary-buckling paths are also disclosed by means of a numerical analysis of equilibrium paths.     

Buckling of thin cylindrical shells: an attempt to resolve a paradox

The classical theory of buckling of axially loaded thin cylindrical shells predicts that the buckling stress is directly proportional to the thickness t, other things being equal. But empirical data show clearly that the buckling stress is actually proportional to t^1.5, other things being equal. As is well known, there is wide scatter in the buckling-stress data, going from one half to twice the mean value for a given ratio R/t. Current theories of shell buckling explain the low buckling stress—in comparison with the classical—and the experimental scatter in terms of “imperfection-sensitive”, non-linear behaviour. But those theories always take the classical analysis of an ideal, perfect shell as their point of reference.Our present principal aim is to explain the observed t^1.5 law. So far as we know, no previous attack has been made on this particular aspect of thin-shell buckling. Our work is thus breaking new ground, and we shall deliberately avoid taking the classical analysis as our starting point.We first point out that experiments on self-weight buckling of open-topped cylindrical shells agree well with the mean experimental data mentioned above; and then we associate those results with a well-defined post-buckling “plateau” in load/deflection space, that is revealed by finite-element studies. This plateau is linked with the appearance of a characteristic “dimple” of a mainly inextensional character in the deformed shell wall. A somewhat similar post-buckling dimple is also found by quite separate finite-element studies when a thin cylindrical shell is loaded axially at an edge by a localised force; and it turns out that such a dimple grows under a more-or-less constant force that is proportional to t^2.5, other things being equal.This 2.5-power law can be explained by analogy with the inversion of a thin spherical shell by an inward-directed force. Thus, the deformation of such a shell is generally inextensional except for a narrow “knuckle” or boundary layer in which the combined local elastic energy of bending and stretching is proportional to t^2.5, other things being equal. Similarly, the modes of deformation in the post-buckling dimples in a cylindrical shell are practically independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell. These ideas lead in turn to an explanation of the t^1.5 law for the post-buckling stress of open-topped cylindrical shells loaded by their own weight.We attribute the absence of experimental scatter in the self-weight buckling of open-topped cylindrical shells to the statical determinacy of the situation, which allows a post-buckling dimple to grow at a well-defined “plateau load”. Conversely, the large experimental scatter in tests on cylinders with closed ends may be attributed to the lack of statical determinacy there.Our paper contains several arguments that are not mathematically water-tight, in contrast to many reports in the field of mechanics of structures. We plead that the problem which we have tackled is so difficult that the only way forward is one of “over-simplification”. We hope that our work will be judged not with respect to its absence of mathematical precision, but by the light which it sheds upon the problem under investigation.     

Upper bounds on plastic strains for elastic-perfectly plastic solids subjected to variable loads

The paper considers shakedown analysis problems for elastic-perfectly plastic solids subjected to quasi-static loads which vary arbitrarily within a given domain. It gives a general inequality which is able to generate Melan's theorem for shakedown, as well as bounds on plastic strains at any point of the solid. These bounds can be made the most stringent by solving a “perturbed” shakedown problem in “finite” or “holonomic” terms. The results presented in this paper are a generalization of those given in a previous paper by the present author[10].     

Wear volume determination during running-in for PEHL contacts

The relation of wear volume and the change of average surface roughness under the “zero-wear” condition was derived, with the assumption that the original profiles of the surface below the wear plane remain exactly the same as before, i.e. no plastic deformation. The flattening of asperities on an engineering rough surface was simulated with numerical techniques. The variation in wear volume and average surface roughness with the depth of wear was studied. The pattern and the correlation length of rough surface were checked and found to have no effect on the relation of wear volume and change of average roughness. The simulated results show that the variation of wear volume and the change of average roughness can be described by a second order polynomial. The model was also validated with experimental results obtained by using a two-disc wear machine.     

Analytical solution for interface stresses due to concentrated surface force

The two-dimensional analytical solution for interface stresses due to concentrated surface force has been deduced, by introducing infinite mirror points which are the images of the load point reflected by the interface and the free surface, and adopting the interchange law of differentiation. The analytical solution can be represented in terms of the summation of the “partial” Goursat's complex stress functions defined in the local coordinate systems with their origins placed at each of the mirror points. It is found that the “partial” stress functions corresponding to a higher order mirror point can be determined from those to the lower one. It is also found that the contribution of the “partial” stress functions to the stress field decreases with the increase of the corresponding mirror point order, therefore, only considering the stress functions corresponding to the first several order mirror points can give the accurate enough solution. Numerical examination by the use of boundary element method has also been carried out to verify the theoretical development.     

On the analysis of sheet metal wrinkling

An analysis for the prediction of wrinkling in curved sheets during metal forming is presented. Using a local approach, similar to that employed for conventional forming limit diagram representations, we construct “wrinkling limit curves” (WLCs) which represent the combinations of the critical principal stresses for wrinkling in curved sheet elements. Wrinkling limit curves are first determined using a bifurcation analysis for plastic buckling in short-wavelength shallow modes. A study of the effects of material properties and sheet geometry on the critical conditions for wrinkling is carried out. We then analyse the effects of geometric imperfections on wrinkling. This analysis is based on the implementation of a finite element scheme. The influence of nonproportional loading is also investigated. In our analysis the material is assumed to be isotropic, elastic-plastic with the plastic part modelled using both J₂ deformation theory and J₂ flow theory of plasticity.     

Stream surface strip element method for simulation of the three-dimensional deformations of plate and strip rolling

A new method—the stream surface strip element method (SSSEM)—for simulating the three-dimensional deformations of plate and strip rolling process is proposed. The rolling deformation zone is divided into a number of stream surface (curved surface) strip elements along metal flow traces, and the stream surface strip elements are mapped into the corresponding plane strip elements for analysis and computation. The longitudinal distributions of the lateral displacement and the altitudinal displacement of metal are constructed respectively to be a quartic curve and a quadratic curve, the transverse distributions of them are expressed as the third-power spline function, and the altitudinal distributions of them are fitted to be a quadratic curve. Based on the flow theory of plastic mechanics, the three-dimensional deformations and stresses of the deformation zone are analyzed and formulated. Compared with the streamline strip element method, the SSSEM considers the uneven distributions of stresses and deformations along altitudinal direction, and realizes an accurate three-dimensional analysis and computation. The simulation examples indicate that the method and the model of this paper are in accord with facts, and provide a new reliable engineering-computation method for the three-dimensional mechanics simulation of plate and strip rolling process.     

Comparison of experimental and theoretical residual stresses in welds: The issue of gauge volume

Welding residual stresses have an effect on many aspects of the integrity of structures but are normally one of the largest unknown stresses. Residual stresses are difficult to measure and to estimate theoretically but are often significant when compared with the service stresses on which they superimpose. High tensile residual stresses can lead to loss of performance in corrosion, fatigue and fracture.In this research, measurement of residual stresses by the neutron diffraction technique is compared to an analysis of a sample geometry by theoretical finite-element procedures. The results indicate good qualitative agreement. One of the key issues in this comparison relates to what is termed “gauge volume” in the measurement technologies and what might be described as a “calculation volume” in theoretical approaches.     

Modelling the evolution of residual stresses during tensile testing of elastoplastic wires subjected to a previous bending operation

During coiling operations high residual stresses are frequently developed in steel wire. In this paper the stress distribution in wires during coiling, unwinding and subsequent tensile testing is modelled for numerous bending degrees, assuming perfect Voce plastic deformation and linear elastic behaviour. The influence of such residual stresses on the observed tensile test data can be deduced. It is shown that coiling with spool radii as used today industrially can lead to measurement of wire properties deviating significantly from the “true” properties of a properly coiled wire. Also, a method is proposed to deduce the original flow behaviour of coiled samples from tensile test curves, hence filtering the effect of the residual stresses.     

Plane strain hot rolling of slab and strip

A numerical solution of the steady-state plane strain hot rolling problem with maximum friction is found in the range from the thick slab to the thin strip rolling geometries. The problem is kinematically determinate, and it is solved on the physical and hodograph planes using the finite-difference approximation of hyperbolic differential equations of the plane strain rigid–plastic flow theory. A sequence of the boundary value problems defined by the slipline field mode is treated as a non-linear vector equation which is solved by Broyden’s secant method. Six versions of the FORTRAN PC programs are written for a wide range of the slab/strip rolling geometries. The solution for the thin strip rolling with large roll radius and small strip thickness reduction is approached to Prandtl’s thin plastic layer with homogeneous velocity field.     

Anisotropic plasticity with application to sheet metals

Hill’s 1948 anisotropic theory of plasticity is extended to include the concept of isotropic–kinematic hardening. The “anomalous” effect can be accounted for by kinematic hardening. It is shown that the quadratic yield function can be used for sheet metals irrespective of its plastic strain ratio R. It is further shown that effects of thickness reduction due to further rolling may be accounted for by kinematic hardening.