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In this paper a numerical procedure for computing the full range of equilibrium thrust lines in masonry arches is presented. According to Heyman, the range can be obtained by shifting the thrust line upwards and downwards until it touches the upper and lower boundary of the arch. This thrust line is computed in such a way as to be the closest one to the geometrical axis. The two limit thrust lines, corresponding to the upper and lower bound of the range, are finally used to compute the degree of safety of an arch through the identification of a domain of equilibrium states and the Heymanian geometrical factor of safety, revisited in the form of a performance factor, that is more practical and immediate to understand.

In the current contribution for the first time a mechanistic model is presented that can be used for analysing and optimising the mechanical performance of straight wall structures in 3D printing processes. The two failure mechanisms considered are elastic buckling and plastic collapse. The model incorporates the most relevant process parameters, which are the printing velocity, the curing characteristics of the printing material, the geometrical features of the printed object, the heterogeneous strength and stiffness properties, the presence of imperfections, and the non-uniform dead weight loading. The sensitivity to elastic buckling and plastic collapse is first explored for three basic configurations, namely i) a free wall, ii) a simply-supported wall and iii) a fully-clamped wall, which are printed under linear or exponentially-decaying curing processes. As demonstrated for the specific case of a rectangular wall lay-out, the design graphs and failure mechanism maps constructed for these basic configurations provide a convenient practical tool for analysing arbitrary wall structures under a broad range of possible printing process parameters. Here, the simply-supported wall results in a lower bound for the wall buckling length, corresponding to global buckling of the complete wall structure, while the fully-clamped wall gives an upper bound, reflecting local buckling of an individual wall. The range of critical buckling lengths defined by these bounds may be further narrowed by the critical wall length for plastic collapse. For an arbitrary wall configuration the critical buckling length and corresponding buckling mode can be accurately predicted by deriving an expression for the non-uniform rotational stiffness provided by the support structure of a buckling wall. This has been elaborated for the specific case of a wall structure characterised by a rectangular lay-out. It is further shown that under the presence of imperfections the buckling response at growing deflection correctly asymptotes towards the bifurcation buckling length of an ideally straight wall. The buckling responses computed for a free wall and a wall structure with a rectangular lay-out turn out to be in good agreement with experimental results of 3D printed concrete wall structures. Hence, the model can be applied to systematically explore the influence of individual printing process parameters on the mechanical performance of particular wall structures, which should lead to clear directions for the optimisation on printing time and material usage. The model may be further utilised as a validation tool for finite element models of wall structures printed under specific process conditions.

Uniaxial damage modelling use constituent components of the stress/strain measured data which have inherent scatter. Models developed for this purpose have also attempted to address the issues related to failure mechanisms within a multiaxial stress state context. This paper presents the multiscale approach to constraint by using the relationship between Monkman-Grant (MG) uniform failure strain and local sub-grain stress state to predict creep damage and rupture under uniaxial/multiaxial conditions. In essence, a global geometric constraint and a time-dependent local sub-grain constraint is defined with the latter controlling the failure response of the geometry. The model compliments and is in agreement with the established NSW model is also used to predict the lower/upper bound of cracking rates in crack dominated geometries. The model is employed into a finite element (FE) to assess its capability to numerically predict the rupture of plain and multiaxial notched bar specimens based on appropriate void growth models. For verification, creep constitutive properties of long term data from uniaxial and multiaxial tests on Grade 92 martensitic steels from various databases, where available, are used to establish the procedure. Given the level of scatter in the data and fabrication and testing uncertainties that cannot be accounted for in such databases, it is shown that the model is sufficiently simple and robust to be developed for use in conservatively predicting very long term failure times.

Bending behaviors can be critical in simulating cables, such as the local effects close to cable end regions or dynamic properties of short cables. Wires form up the cable and tend to slip relative to each other under bending load. Complex interactions among wires make the modeling of cables challenging. In this work, we propose an analytical model for bending parallel wire cables in which the cable cross section with layered wires is idealized as a composite section of the laminated beam. The slips and interactive forces among wires are modeled by the interfacial slips and the resulting interactive forces at the interfaces of the composite section. In this way, the bending cable can transform into an in-plane stress state laminated beam. The state space method is then applied to solve the problem, and the numerical results agree well with the experimental results and analytical formulations in the literature. Bending stiffness of the cable differs from lower bound to upper bound according to the anti-slip ability among wires, from full-slip state to no-slip state. The overall deflection, as well as the local deformations and stresses of the bending cable, can be simultaneously obtained. These results show that the present model is efficient and accurate in simulating bending cables, which is capable of extending to more cable studies.

Different plastic collapse modes may have different effects on the overall behaviour and load-bearing capacity of an elastic–plastic structure subjected to variable loads, and they may even be determined by different material plastic constants (for general plastic hardening materials). Both lower bound static and upper bound reduced kinematic approaches have been implemented with appropriate finite element realizations and mathematical programming techniques to study the nonshakedown modes for elastic plastic bodies under cyclic loads. For sufficiently complex structure and loading program, it has been firstly demonstrated that an elastic–plastic body may fail by rotating plasticity collapse rather than the simpler alternating plasticity one among other possible modes. That and other results also lead to interesting problems for further studies.

A theoretical model based on upper bound approach is proposed to study forging penetration in a radial forging process. Using the transformations for both the geometric coordinate system and the velocity field, the parabolic discontinuity boundaries in axisymmetric problem are mapped into straight lines in the transformed field. As a result, the present model is simplified and its implementation is analogous to the rigid block upper bound approach in plane strain problem. In order to quantify the penetration of plastic deformation, forging penetration depth is introduced and defined as the radial distance from the outside surface of the workpiece to the intersection point of the assumed plastic and rigid regions. This model is verified by comparing the predicted forging load with published experimental data, and by comparing the predicted forging penetration depth with that of the finite element simulation. Finally the effects of process parameters such as the radial reduction rate and the inlet angle of hammer on the forging penetration are investigated.

The paper aims to perform shakedown analysis of truss structures with nonlinear kinematic hardening. Both the static and kinematic shakedown analyses of truss structures were conducted analytically and numerically to bound the shakedown limit. First, the problem statement leads to the lower bound (primal) formulation accounting for nonlinear kinematic hardening extended from the Melan׳s static shakedown theorem. In particular, the Holder inequality is then utilized to establish the corresponding upper bound (dual) formulation from the lower bound formulation as well as to confirm the duality relationship between them. The derived upper bound formulation is an equivalent form of the Koiter’s kinematic shakedown formulation for trusses without involving time integrals. Further, both the primal and the dual analyses of truss structures were conducted using the optimization algorithms provided by MATLAB. Accordingly, the primal analysis and the dual analysis are validated by converging to the shakedown limit efficiently. Finally, the step-by-step finite-element analysis by using ABAQUS is also performed to verify the analytical formulation and numerical implementation.

Talbert and Avitzure introduced the implementation of a quadratic stream function for derivation of velocity field governing through streamlined die extrusion. In the present study, a non-quadratic stream function is introduced to model various streamlined die processes. In the modified formulation an exponent was designated to the stream function in order to obtain a generalized model. The quadratic stream function approach was restricted to processes with axial flow direction. In contrast, the modified model is capable to study the common forming processes i.e. simple and bimetal extrusion and drawing. The actual velocity field was derived from the generalized field by upper bound technique. The plastic flow was studied through distorted bands and elements that obtained by Eulerian description of motion. Subsequently, the modeling of extrusion and drawing proceed by the calculation of the effective plastic strain distribution. Streamlined die has been designed and proposed based on the fact that it requires less external power than the conventional operations and that deforming material encounters smoother plastic flow. The results revealed the length of die highly influences the deformation mode by affecting the shear component of strain. The capability of the modified stream function to capture various metal forming operations with streamlined die was investigated.

In this investigation, an upper bound solution is presented for the multistage forming process to produce polygonal cross-section thick tubes by means of a roll forming rig with flat rolls. The modeling and simulation is presented so that it can be generalized for arbitrary number of sides and stages. Comparison of theoretical and experimental results for tube cross-section dimensions (height, wall thickness and outer corner radius) show good agreement for a square section tube and demonstrated the capabilities of the new formulation presented.

In this paper, an upper bound solution is used to determine the ring rolling power and force. An admissible velocity field and strain rates are derived from the parametric definition of streamlines in the deforming zone. Minimizing the upper bound power with respect to neutral point position, the neutral point situation and the rolling force are determined. The present method is validated using experimental results extracted from ring rolling mill. Moreover, using ABAQUS/Explicit software, the ring rolling process was simulated in every respect. Comparison of the present upper bound results with those from experimental study, finite element simulation and our previous slab analysis, have proved the accuracy of the present analysis. As compared with the FEM simulation, this method is very much quicker and less expensive and could be used as an engineering tool in the ring rolling industrial applications.

This paper provides a new generalized formulation based on the upper bound method for the three-dimensional analysis of the forward extrusion of non-symmetrical sections. The proposed method has been formulated in such a way as to present a more realistic definition of deforming region by eliminating internal discontinuities and also velocity discontinuities at the entry and exit surfaces which are defined by three-dimensional curved surfaces. Using this new formulation, extrusion of shapes such as L-shape, rectangle, hexagon and a spur gear was analyzed. The distribution of velocity and strain, three-dimensional material flow as well as extrusion load were obtained. Finite element analysis for the same problem was also carried out and the comparison of the results showed the capability of the present method. In fact, the results of the analytical velocity and strain distributions are in close agreement with those given by the simulation in terms of magnitude and distribution. Influence of the process parameters such as extrusion ratio and aspect ratio on the strain distributions was investigated. Comparison between theoretical and experimental data was carried out and close agreement was observed. Also, improvements of the results of the present method were observed comparing with previous works.

There is a trend towards component miniaturization and strong drive towards cost effective and sustainable metal forming techniques of miniaturized components. This paper presents an upper bound solution for the optimization of open-die forging/extrusion processes in the forming of micro-pins from a sheet metal. Using such an analytical modeling approach, the critical blank thickness, the resulting final part geometry, together with the required forming load were predicted based on the location of the neutral plane under the punch during the process. Based on the phenomenological findings of the process, the geometry size factor, x, was introduced explaining its relative importance to the model. Experimental results obtained from C11000 copper samples using a progressive microforming process was found to agree well with the results predicted by the model. The results were also validated with other results reported before from a similar process.

The dynamic plastic response of a simply supported circular plate is analysed. Emphasis is given to the plate behaviour after it has broken free from the supports due to a local material failure. The theoretical rigid plastic analysis predicts various features of the response such as the time to failure, residual kinetic energy and the critical velocity at failure. The residual kinetic energy of the plate could be significant enough to cause secondary impact damage. It is shown that the shape of the plate changes after breaking free from the supports, which is important for forensic investigations. The solution for various cases were proven to be exact in the context of the upper and lower bounds theorems of the theory of plasticity.

This paper analytically investigates plastic limit pressure of orthotropic strain-hardening cylindrical vessels under internal pressure. It is an interesting problem to illustrate the interaction between strengthening and weakening behavior during the deformation process. The Voce hardening law and Hill's yield criterion were adopted in the paper. A sequence of static and kinematic limit analysis problems were performed by updating the yield criterion and the deformed configuration. The equality relation between the greatest lower bound and the least upper bound was confirmed explicitly. Accordingly, exact solutions of plastic limit pressure were developed with an integral term. Particularly, exact closed-form solutions were obtained for certain values of the hardening exponent of the Voce hardening law. Finally, numerical efforts were also made for rigorous validations.

Reshaping process based on the cold roll forming technology is a process in which noncircular pipes can be produced from circular pipes. A generalized upper bound solution for the deformation of thick tubes between four flat rolls has been formulated. The deforming zone has been divided into two areas namely the “contact region” and the “free region”. Owing to the different physical character of these deforming zones two different kinds of admissible velocity fields have been obtained for each of the deforming zones. To formulate the stream lines and stream surfaces in the deforming zones use has been made of the Bezier curves which have been utilized to define stream lines in such a way that could be manipulated easily to obtain the optimum shape for the upper bound solution to be optimized. This new formulation was used to predict the upper bound on power. Using the theory presented here, the influence of process parameters such as radius of rolls, initial pipe dimensions; amount of roll gap reduction and the roll speed on the final rolled product were investigated. In order to verify the theoretical results, an experimental rolling rig was designed and built which comprised of four flat rolls. Using this rig Al 6101 round pipes were rolled and the variations in the above-mentioned parameters were investigated. Quantities such as energy, wall thickness, and corner radius of the tube were observed and measured. Comparison of theoretical and experimental results showed good agreement and hence demonstrated the capabilities of the new formulation presented here.

An adaptable die is one that not only produces the correct geometrical shape, but also is designed through an adaptable method to impart other desirable properties to the product or process. In this first part of a series of papers, six kinematically admissible velocity fields are developed for use in upper bound models for axisymmetric extrusion through various dies, including extrusion through adaptable dies. Three base velocity fields are presented: (1) assuming proportional angles in the deformation zone, (2) assuming proportional areas in the deformation zone, or (3) assuming proportional distances from the centerline in the deformation zone. The base velocity is modified by an additional term comprised of two functions. One function allows extra flexibility in the radial direction, and the second function allows extra flexibility in the angular direction. There are two forms of the second function, which meet the required boundary conditions. The flexibility function in the radial direction is represented by a series of Legendre polynomials, which are orthogonal over the deformation region. The power terms derived for these velocity fields for use in upper bound models are also presented. Part 2 of this series compares the results obtained in upper bound models for the six velocity fields for a spherical extrusion die. In Part 3, the use of the best velocity field for extrusion through streamlined dies is developed to determine the adaptable die shape, which minimizes the required extrusion pressure. Additionally, the adaptable die shape is compared with results from Yang and Han for arbitrarily curved and streamlined dies.

Radial forging is an open die forging process used for reducing diameter of shafts, tubes, stepped shafts and axels, and creating the internal profiles for tubes such as rifling the gun barrels. In the present research, a new model based on calculating the deformation work was developed to find an upper bound limit for the deformation load in the case of radial forging of rods and tubes. Also, the model was used to assess the effects of the process parameters. The accuracy of the model was tested by comparing the predicted results with those achieved from the experiment at work of Uhlig [Investigation of the motions and the forces in radial swaging. Doctoral Dissertation, Technical University Hannover, 1964]. A good agreement was found between the two sets of results.

An upper-bound analysis of the pressure-assisted injection forging (PAIF) of thick-walled tubular components with hollow flanges was conducted to predict the maximum forming-pressure requirements. The polymeric pressurizing material and the pure aluminium work material employed were assumed to obey the von-Mises' yield criterion, the two materials being divided into three and six zones, respectively, and described using kinematically admissible velocity fields. An upper-bound solution was derived based on this model. Experiments were conducted to validate the solution. The results suggest that the upper-bound analysis has sufficient accuracy for the prediction of the forming-pressure requirements. Subsequently, a parametric analysis was conducted to study the effects of variation of the principal parameters of the process and the geometry, such as tube thicknesses, flange dimensions and friction, on the forming-force requirements, which showed other useful applications of the upper-bound solutions.

A lower bound to the creep rupture time of internally pressurised thick cylinders is derived. Material behaviour is described by a phenomenological creep rupture theory that accounts for all phases of creep, and for the full coupling between the deformation and damage processes. To obtain the desirable lower bound, the effective stress and the equivalent rupture stress, which represent the effects of multiaxial stress states on the creep strain and damage rates, respectively, were approximated by the maximum shear stress in the constitutive equations. By comparing the lower bound estimations for a wide range of cylinder dimensions and different engineering materials with the rupture times determined from accurate finite element calculations, it is shown that the lower bound estimations provide quite conservative lifetime predictions.

The upper bound approach is utilized to analyze the forward extrusion of metals through a rotating conical die. A continuous spherical velocity field, compounded with a torsional velocity field constrained in a conical channel, is assumed. The circumferential slippage between the material and the tooling is analyzed by carrying out a minimization procedure on the expression derived for extrusion pressure. The length of material twisting inside the container is evaluated both using a slab method and an approach based on the balance of moments. This enables a complete upper bound estimate of extrusion pressure to be obtained. The reasonably simple model thus produced enables an analytical solution to be generated that agrees well with the experimental results.

Precision forging is an important manufacturing procedure of spline and spur gear forms. It has advantages of improved strength, good tolerance, saving billet material, dispensing with the cutting, etc. In this paper, a mathematical model using an upper bound method is proposed for forging of spur gear forms and spline to investigate the plastic deformation behavior of billet within the die cavity. The material of solid billet was assumed as rigid–plastic and the shape of the tooth profile was accounted for the mathematical modeling of the kinematically admissible velocity field assumed for the plastic zone. The non-uniform velocity was employed for simulating the inhomogeneous deformation and the effect of barreling during the forging. Using the present model, various effects of forming parameter such as the friction factor, reduction, number of teeth, etc. upon the non-dimensional forging pressure, forging force and barreling of the spur gear forms and spline were analyzed systematically and the results compared with those of other researcher's analytical and experimental work. It is shown that the present modeling of the process improves knowledge of the process design performance for the precision forging of spur gear form and spline.

A new upper bound method for the analysis of three-dimensional deformation in the flat rolling of bars is proposed. In the conventional upper bound method, stress distribution is not calculated. Hence, a new method of calculating stress distribution is proposed in which the hydrostatic stress in each element and the contact stress on the contact surface between material and roll in each element are calculated. In the conventional upper bound method, the structure of the computer program also depends highly on the kinematically admissible velocity fields assumed. Hence, a new method of analysis in which the structure of the computer program depends minimally on the kinematically admissible velocity fields assumed is proposed. In the analysis of strip rolling, the calculated roll force and roll torque agree with the roll force and roll torque calculated from Sims’ slab method. In the analysis of the flat rolling of bars, the calculated width spread agrees with the width spread obtained by experiments reported in the literature. The effects of a reduction in thickness, roll radius, material width-to-height ratio, front tension, back tension, and front and back tensions on the width spread, forward slip, roll force and roll torque are demonstrated. Hence, the validity of the new upper bound method is confirmed.

Among the many factors that affect the dimensional accuracy of the closed-die forging of a crown gear form, an important factor is well designed shape of the initial billet or preform, as this can markedly improve the dimensional accuracy of the forged product. The paper presents kinematically admissible velocity fields for closed-die forging of the crown gear form which take into account the profiled teeth chosen kinematically by approximating these as trapezoidal teeth and then simulates using the velocity fields to obtain upper bound to the load and the deformed configurations of the various shaped solid billets or preforms. Based on these simulations, the experimental investigations were carried out to obtain a forged crown gear form with the good quality, using two types of model materials, lead and plasticine. The preform shape which improved the dimensional accuracy of the forged crown gear form is presented and the results commented upon.

In analytical method based on the upper-bound theory is proposed to investigate the three-dimensional off-centric extrusion of arbitrarily shaped sections from arbitrarily shaped billets through linearly converging and smooth curved dies. A set of generalised kinematically admissible velocity fields are derived on the assumption of Bezier-type streamlines that provide compatibility of surface representation with most CAD/CAM systems. To obtain a more realistic non-uniform metal flow, a special velocity function was also incorporated into the derived velocity fields and work hardening effect of the material considered. Based on the method presented, for a given reduction in area, material property, friction condition and off-centric positioning of the exit cross-section, predictions of the deforming grid pattern, curvature of the extruded product as well as upper-bound to the extrusion pressures may be obtained. The above procedure is highlighted and commented upon.

In this paper, the forging of helical gears has been investigated. A punch and a die are tooth-shaped for the guiding type forging of helical gears. The punch compresses a cylindrical billet placed in a die insert. As a consequence the material of a billet flows into the tooth region. The forging has been analysed by using the upper-bound method. Kinematically admissible velocity fields have been developed, where an involute curve has been introduced to represent tooth profile of the gear. Numerical calculations have been carried out to investigate the effects of various parameters, such as modules, numbers of teeth, helix angles and friction factors on the forging of helical gears. Some forging experiments were carried out with commercial aluminium alloy to show the validity of the analysis. Good agreements were found between the predicted values of the forging load and those obtained from the experimental results.

A kinematic method is developed to evaluate the shakedown limits for plates subjected to variable (quasistatic as well as dynamic) loads. The shakedown kinematic theorem is transformed into a reduced form, to which the upper-bound method involving construction of kinematically admissible mechanisms with yield curves used in plastic limit analysis can be applied. As an example, inadaptation of a circular plate clamped at the edge under variable uniform loading is considered.

The initial collapse behavior of elliptical tubes subjected to lateral compression is investigated. Both unbraced and braced tubes are analysed, and the main factors which affect the initial collapse behavior are discussed and interpreted. Lower and upper bound theorems are applied to bracket the collapse load in a direct manner. The initial collapse loads for braced circular tubes, obtained as a special case of elliptical tubes, are compared to previous research results. The results and concepts presented herein can be useful in the development of design criteria for energy dissipating devices. Elliptical shapes, when loaded parallel to their major axes, can have an advantage over their circular counterparts in that they possess larger collapse strokes and an increased energy dissipation potential per unit mass.

In the present study the SERR technique has been reformulated to deal with extrusion of sections with/without re-entrant corners. A comprehensive computational model has been developed for this reformulated technique. The model has been used to analyse extrusion of I-section bars and the computed results have been compared with experimental values available in literature.

In this paper, the analysis of plane strain forward extrusion through wedge shaped dies is implemented by using a finite element procedure based on a dual (upper bound) formulation. Since the von Mises yield criterion is employed and expressed in an inequality form, stresses do not appear in this upper bound formulation. As a result, neither complicated stress updating nor rigid zone treatment is needed in this approach. The effect of material strain hardening can be incorporated in the analysis by updating local yield stresses stepwise with deformation history. The computed results are first compared with existing analytical solutions in good agreement. Effects of friction, die semiangle, height reduction ratio and material strain hardening on the extrusion load of aluminum are then investigated.

Numerical method of construction of axi-symmetric slip-line fields and associated velocity fields in the form of computographic plots of axial and radial velocity components, earlier given by the authors for rod extrusion and tube extrusion, was extended to the solution of two types of combined rod and tube extrusions: (i) where the rod and tube are extruded in opposite direction to each other and (ii) where, both the rod and tube move in the same axial direction as the punch movement during the extrusion process. Computographic plots of slip-line fields and associated velocity fields in the (r, z) plane are given and the deficiencies noticed in the solutions are highlighted. The results of specific punch pressures are observed to compare well with those estimated, using simple upper bound analysis and the experimental work of the authors.

The paper gives a new method no determine the mean unit contact pressures on a material—tool contact surface in cross wedge rolling processes (CWR). The dependencies worked out on the basis of the energy and the upper bound methods permits rolling forces to be determined which are comparable to experimentally measured ones. The analysis provides equations which relate the mean contact pressure q m to the basic process parameters, namely the forming angle α , the spreading angle β , the relative reduction of a portion δ and the shear friction factors m and m k .

The paper presents the results of an experimental investigation carried out in quasi-static incremental upset-forging of spur gear forms around the periphery of initially solid cylindrical billets, the outer radius of each of which was of the same size as the radius of the dedendum circle of the spur gear to be forged. The punch or die used was of the same size and shape and had the same number of teeth at its circumference as that in the female stationary die and corresponds to the shape and size of the gear to be forged in each case. For the analysis an upper bound elemental technique previously applied to spline forging was first modified for the tapered teeth and was subsequently adopted to the analysis of forging spur gear forms having the actual profile teeth around its circumference each with the pressure angle of 14½°. The results of theoretical investigations presented here are confirmed with experiments and are compared with the existing upper bound analysis.

A simple kinematically admissible velocity field for three-dimensional deformation in near-net shaped forging of a gear coupling form is analysed which takes account of the profiled teeth chosen kinematically by approximating these as straight taper teeth. The upper bound load and the deformed configurations are predicted by the velocity field at varying punch movements considering differing frictional factors. Experiments were carried out on specimens made of a model material, plasticine at room temperature where talcum powder was used as a lubricant. The theoretical predictions of the forging load and the dimensionless forging pressures are found to be in reasonably good agreement with the experimental results.

Closed-die forging of external spline gear forms is analysed using an upper bound elemental energy method. The kinematically admissible velocity field for the three-dimensional deformation was obtained on the assumption that the profiled surface of the tooth was straight but tapered. The upper bound to the deforming load necessary and the deformed configurations are predicted using integration of the formulation of energy expressions which were obtained from the presented velocity fields. Two different forging methods denoted here as A, viz. a side extrusion method and B, an upset forging method, the details of which are given in the text, were considered in the present analysis and the theoretical results compared with experimental ones. Experiments were carried out on plasticine as model material at room temperature where talcum powder was used as a lubricant. The present investigation revealed that the analytical method B predicts a reduced forging load and improved configuration better than method A for the forged products.

The progressive deformation of axi-symmetric cylindrical rods of different initial free lengths to diameter ratios and materials into shaped heads: triangular, square, hexagonal and octagonal shapes is simulated using the stress equilibrium technique based on slab method of analysis. The mechanics of the heading process, before the flash sets-in, in each case, is analysed in two stages: upset forging and filling of the shaped die cavities. The theoretical results estimated compare very well with the experimental and those calculated previously by the authors using the upper bound elemental technique

A simple analysis based on an upper bound elemental technique is presented for the incremental forging of a near-net shaped splined rotor shaft-type component formed using a cylindrical rod-type specimen. The results of an experimental investigation made into the working pressures and deformation modes observed during progressive forging of tellurium-lead specimens of different aspect (length to diameter) ratios into spline shaped components are compared with those predicted theoretically. The agreement was found to be reasonable and encouraging.

The interference effect of double symmetrical cracks of either equal or unequal crack depth on the plane-strain general yield strength under tension is analyzed based on the upper and lower bound theorems. Theoretical results are compared with experimental results. The experimentally obtained range of crack pitch as influenced by the interference effect can be predicted accurately by the theory. The discrepancy is recognized at each crack pitch, especially for the single symmetrical crack, between the experimental value of the ratio of mean tensile stress in the net section at general yielding to the lower yield point and the constraint factor calculated from the theory. Strain figures obtained by Fry's etching method show that plastic deformation occurs approximately along the line of velocity discontinuity employed in the upper bound considerations.

The technique of statically admissible plastic discontinuous stress fields is used in the plastic design of machine parts of complex shape. If the shape of the part is given the stress field is used to find a lower bound on the bearing capacity. In the case of plastic design an appropriate stress field gives a safe estimate of the shape of the designed part. Complex three-dimensional stress fields are composed of elementary sub-fields, all geometrical and statical parameters of which are tabulated in order to simplify the design procedure. Thus the procedure consists in appropriate arrangement of sub-fields into a complex system of stresses.

A new technique, the ‘end-fixity coefficients method’, is developed for the prediction of upper- and lower-bound loads to cause the plastic collapse of thin, uniformly loaded rectangular plates of rigid-perfectly plastic material, which is assumed to flow according to the Johansen yield criterion, re-deriving some well-known results and extending the analysis to other cases of loaded plates. The proposed theory has unified application for the analysis of plate bending, under various boundary conditions, taking into consideration the clamping conditions of plate sides.

Previously postulated, lower bound estimates of the buckling loads for axially loaded unstiffened cylinders are simplified by means of a Donnell-type approximation to provide compact, explicit, analytical expressions which could prove particularly suited to design. For a wide range of practical geometric parameters these simplified expressions are shown to provide close approximations of the exact lower bound loads and associated modes. In addition they allow the independent and significant influence of length to radius, radius to thickness, and Poisson's ratio to be isolated, and suggest a convenient means of summarising buckling loads in terms of a single composite geometric parameter.

An upper-bound solution for the analysis of hydrostatic bulging is found by minimizing the plastic energy consumption rate divided by the fluid volume change rate. The corresponding upper-bound pressure is found through optimization with respect to some parameters given in the assumed velocity field. Computations are carried out for axisymmetric hydrostatic bulging of isotropic work-hardening sheet metal. The computed results are then compared with those by the finite-element method and by the finite-difference method as well as the reported experimental results. The comparison shows that the present method renders excellent predictions in pressure and strain distributions with much reduced computational time, which are still in close agreement with those obtained by the numerical methods.

An investigation of the usefulness of the Taylor theory of ductile hole formation for the failure of metal targets shows that the model is a good lower bound on the work done provided a realistic flow stress is chosen for the target material; the use of the stress at a natural strain of 1·0 is found to give consistent results. It is observed, however, that strains in perforated targets are vastly different to expectations from the Taylor model. The Thomson model for dishing failure of thin plates predicts energies much lower than those observed. The inclusion of the work done in bending into the Thomson solution improves the energy calculation and gives an explanation for the transition from dishing to ductile hole formation as target thickness to projectile diameter ratio increases.

The classical Bree problem—which represents an uniaxial model of a thin tube subjected to combined internal pressure and cyclic thermal stress across its wall—can be simulated by means of the pulley test in which a wire or strip specimen is subjected to combined steady tensile stress and cyclic bending stress. In this paper, accumulation of ratchet strain in the pulley test is investigated using a linear kinematic hardening material model from which perfect plasticity can be generated as a special case. The results of the investigation show that asymptotic ratchet strains are linearly related to the excess in mean stress σ D above its value σ∗ D at the ratchetting limit regardless of the thermal stress amplitude. Comparisons with test results on copper wire specimens—which exhibit non-linear hardening rate—confirm the qualitative validity of this simple relation . Deviations between theory and experiment are attributed to metallic cyclic creep. Further, perfect plasticity results are shown to be well predicted by a linearized lower bound estimate.

Both upper and lower bounds on final displacements of dynamically loaded, rigid-plastic structures can be easily calculated. These bounds are obtained from an assumed, statically admissible stress field and a kinematically admissible velocity field, respectively. The range between these bounds is a function of the difference between stresses in these assumed fields and the actual stress distribution in the structure. This paper identifies the manner in which structural characteristics and the load distribution influence accuracy of the bounds. Generally, the range between the bounds is small for one-dimensional deformations in response to uniformly distributed loads and larger for two-dimensional deformations and more concentrated loads.

A novel mode of deformation of circular tubes axially compressed on to a die is examined experimentally with regard to its energy absorbing capacity. The tube splits as a number of axial cracks are produced and the strips so formed are bent into curls. The major part of the deformation takes place at a nearly constant load which can be increased by preventing the formation of the curls. A simple upper bound approach is used to estimate this operating load. Compared with axial buckling and the inversion mode, the operating load for a splitting tube is somewhat lower. However, the fact that the stroke is of the order of 90% of the tube length means that the performance as measured by the total energy absorbed is often as good as these two efficient modes of deformation.

Constitutive relationships are discussed for materials which undergo creep rupture due to cyclic loading. Relationships are proposed which describe the strain rates in terms of the current stress and a single state variable. An approximate method is derived which enables a lower bound on the rupture life to be obtained for kinematically determinate structures subjected to cyclic load and isothermal conditions. The bound on the rupture life is expressed in terms of the energy dissipation rates within the structure corresponding to stationary-state creep. The effect of multi-axial stress creep-rupture upon the structural performance is examined and bounds are derived for materials which obey maximum tension and octahedral shear stress criteria. For both multi-axial stress rupture laws and structures subjected to cyclic and reverse load conditions formulae are derived which express the lower bound rupture life in terms of the behaviour of a steady-load uni-axial creep rupture test. Results of experiments which have been carried out on copper and aluminium beams are presented for cyclic and reverse load conditions. For both rupture laws the experimental rupture times are closely predicted by the representative rupture stresses and uni-axial data.

Deformation by 〈111〉-pencil glide has been analyzed by an upper-bound model which combines a least-shear analysis and Piehler's maximum virtual work analysis. The least-shear analysis gives exact solutions if three 〈111〉 slip systems are active, while the maximum work analysis provides solutions for the case of four active slip systems. Independent checks are used to determine which solution method is appropriate. Computer calculations using this model have been made to determine; (1) the orientation dependence of the Taylor factor for axisymmetric deformation; (2) the yield loci for textured materials having [100], [110] and [111] sheet metals and rotational symmetry; (3) the isotropic yield locus for randomly oriented materials; and (4) flow stresses along critical loading paths for various assumed textures with rotational symmetry. The latter calculations indicate that anisotropic yield loci of textured bcc metals with rotational symmetry are much better approximated by σ x a + σ y a + R¦σ x − σ y ¦ a = (R + 1)Y a where R is the strain ratio and Y is the tensile yield strength with an exponent a = 6 rather than with a = 2 as postulated by Hill. It is not known how well upper-bound calculations like these represent actual yielding behavior.

Hollow cylinders of initial height to diameter ratios between 1:1 and 2:1, with central holes of various diameters, were compressed axially under dry, quasi-static conditions. In the initial phase of compression three modes of flow could be identified, distinguished by single (I), double (II) or triple (III) barrels on the free surface. During continued compression, mode III degenerated into an unstable process similar to the collapse of thin-walled cans, whilst specimens initially displaying mode II changed to mode I or retained their initial form. Those cylinders that initially displayed mode I were found to retain this form throughout their subsequent compression. The deformation characteristics outlined above are shown to be dependent on the initial height to diameter ratio and the size of the bore. A preliminary axi-symmetric upper bound analysis is presented, but the simple models used are shown to be insufficient to represent accurately the complex phenomena observed. It is suggested that finite-element elastic-plastic analysis may be more appropriate.

An analysis of the upset forging of polymeric rings, which obey the pressure-modified von Mises yield criterion and associated flow rule, is presented using the upper bound method. A simple velocity field, consisting of an incompressible and a compressible part, is assumed. The effects of pressure-dependency of the materials on forging pressure, bulging, location of neutral radius and calibration curves for various frictions and geometries are assessed.

The plane-strain general yielding of plates with deep single notches is studied using slip-line field analysis, Rice's upper bound solution and finite element analysis. When the plates are subjected to arbitrary combinations of axial force and bending moment, it is possible to calculate the corresponding constraint factors and to map them on the constraint factor plane. It is shown that there is a change in the mode of general yielding from that predicted by slip-line field analysis to that predicted by Rice's upper bound solution. This mode transition occurs when the axial force becomes dominant in the first and third quadrants of the constraint factor plane. The transition is confirmed by finite element analysis and also experiments on notched steel plates.