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The advent of electronic computation had a profound effect on many aspects of present life and, obviously, structural mechanics also was affected. The impact was so significant that some researchers predicted a sort of revolution, the advent of a “discrete” mechanics that would replace the traditional continuum formulation. A few years have elapsed since then and we can state that that such a revolution did not occur as yet. Nevertheless, structural mechanics did not remain unaltered: computations gave rise to a critical review of some aspects which certainly did not alter the foundations of the field, but contributed to clarify some points and improve the overall picture under some respect.

Riassunto. – Si illustrano i problemi statici sollevati dalla costruzione del Duomo di Milano e le soluzioni adottate per risolverli, sottolineando le similarità e le differenze che sussistono rispetto alle grandi cattedrali gotiche d’oltralpe. Il Duomo di Milano è stato concepito come una cattedrale del nord Europa, ma la sua costruzione è stata affidata prevalentemente a maestranze italiane, che avevano una abbastanza limitata confidenza con gli aspetti tecnici di questi monumenti. Di conseguenza, il Duomo di Milano si è avvalso solo parzialmente delle soluzioni tipiche delle cattedrali gotiche d’oltralpe e in molti casi sono state concepite soluzioni, il che fa del Duomo un monumento sotto molti aspetti unico anche dal punto di vista stutturale.***Abstract. – The static problems raised by the construction of the cathedral of Milan and the structural solutions envisaged to face them are discussed, pointing out similarities and differences with respect to the great gothic monuments of northern Europe. Milan's cathedral was conceived as a northern gothic church, but its construction was carried on by Italian masons, little acquainted with the technical aspects of such monuments. As a consequence, northern gothic solutions were only partially adopted and completely new ones were worked up, so that the cathedral developed into a monument under several respects unique also from the structural point of view.

The collapse behavior of cylindrical shells pressurized from outside is examined. Attention is focused on tubes of moderate thickness, as required by very deep water pipelines or some innovative nuclear power plant proposals. Their collapse is expected to be dominated by yielding but, because of the decreasing nature of the post-collapse evolution, interaction with instability is likely to be significant enough to demand consideration. At present, no quantitative assessment of such effect is available, because little study has been devoted to tubes in this thickness range. Plasticity–instability interaction is activated by imperfections and to assess their influence on a systematic numerical study is undertaken. Computations produce a meaningful measure of the collapse pressure and it is proposed that the allowable pressure be determined on its basis, by introducing a suitable safety factor. This is chosen so that results reproduce those provided by presently accepted procedures in the well explored and reliable range of medium-thin tubes. When the same factor is applied to thicker tubes, the resulting allowable pressure is significantly higher than the values suggested by codes, which apparently react to the present lack of knowledge by assuming an extremely conservative attitude.

A formulation for finite element limit analysis of a certain class of 3D perfectly plastic solids governed by von Mises’ plasticity condition is presented. A planar symmetry constraint for both geometry and displacement field is assumed to analyze plane problems where the variable nature of transverse dissipation must be considered. A mixed locking free and low distortion sensitive element is formulated on the basis of the natural approach. The solution procedure exploits the kinematic theorem of limit analysis, cast in the form of a minimum problem for a convex but non-smooth dissipation functional. Applications to a notched specimen and to a bolted joint are presented to stress the importance of transverse effects in some problems commonly modeled as purely 2D.

The imperfection sensitivity of cylindrical shells under external pressure is studied, focusing attention on tubes of considerable thickness, such as those to be employed for very deep water pipe lines or in some innovative nuclear power plant proposals. In such tubes collapse is dominated by yielding, but the decreasing nature of the post-collapse evolution makes the effects of interaction with instability significant enough to demand consideration. On the other hand, available theories assessing the imperfection sensitivity of thin shells do not apply in this context and empirical design formulas used for the medium-thick tubes typical of present oil industry applications are extremely conservative. The interaction phenomenon becomes inherently different as thickness increases, and its correct description appears a must in view of a rational assessment of the load bearing capacity. This paper is intended as a first step in this direction.

This paper provides the theoretical collapse loads of thick, long cylindrical shells subject to pressure and axial forces. Tubes are made of isotropic, perfectly plastic von Mises' material. Axial strains are assumed to be constant but possibly different from zero, so that elongation is permitted. This assumption, together with axial symmetry and the isochoric nature of plastic flow, unambiguously defines the set of possible collapse mechanisms, and collapse loads are computed on this basis. Results are contrasted to those presently available, based on thin-shell assumptions. Comparison shows that differences are of engineering significance, well worth considering for thick tubes, such as those envisaged in some nuclear power plant applications.

The numerical solution of the limit analysis problem has experienced a growing interest in recent years. Methods developed in this context can be employed also to obtain indications on the structural response subsequent to collapse, which is required in several situations, such as for shells employed as shock absorbers or energy dissipators. The procedure is known as sequential limit analysis and, as its name suggests, is based on a sequence of limit analysis solutions referring to progressively updated configurations. In this paper, the limit analysis procedure proposed by Capsoni and Corradi [Int. J. Numer. Meth. Eng. 40 (1997) 2063] is employed to this purpose in conjunction with the TRIC shell element developed by Argyris and co-workers [Comp. Meth. Appl. Mech. Eng. 145 (1997) 11], which is modified to some extent to adapt to the rigid-plastic context. Some examples show the effectiveness and the accuracy of the method, which compares well with results obtained from complete, although computationally demanding, incremental elastic-plastic approaches.

The limit analysis problem for plates in bending is considered. The failure criterion for the material is assumed as orthotropic, with possible non-symmetric strength properties. According to Kirchhoff’s hypothesis, the plate is conceived as a superposition of layers, individually in plane stress situation, and continuity is enforced by means of a kinematic assumption. By exploiting previous results, recently established by the authors, the expression of the dissipation power per unit plate area is defined on this basis and the kinematic (upper bound) theorem of limit analysis is cast in a form suitable for numerical solutions. To this purpose, efficient algorithms successfully employed in the isotropic case can be used with minor modifications. The effectiveness of the procedure is demonstrated by solving some homogeneous plate examples. Results permit the assessment of the influence of different aspects, such as the ratio between strengths along the orthotropy directions, the tensile to compressive strength differential and the inclination of the orthotropy axes with respect to the sides. The effects of in-plane edge constraints are also discussed and it appears that they are emphasized considerably by anisotropy. Even if referred to specific cases, some conclusions can be regarded as fairly general.

The knowledge of the post-collapse response of structures is required in several situations. Typical examples are shells used as energy absorbers or bumpers, which must be able to undergo large plastic deformations by dissipating a sufficient amount of energy. The procedure known as sequential limit analysis can be employed to this purpose. The evolution of the structure after collapse is tracked by a sequence of rigid–plastic limit analyses, with the structural geometry progressively updated on the basis of the collapse mechanism computed in the previous step. A number of methods have been developed, mostly in recent years, for the finite element solution of the limit analysis problem. In this paper, a procedure proposed by one of the authors is employed, which a rather extensive computational experience proved reliable, stable and accurate. The procedure demands that the finite element be formulated on the basis of the natural approach, which suggests the use of a simple but well performing triangular shell element, named TRIC, recently developed within this framework by Argyris and co-workers. The formulation has to be modified to some extent to adapt to the rigid–plastic context, but the quality of the element performances is maintained. Some computations, referring to plates and shells, are presented, which assess that the “marriage” between the TRIC element and sequential limit analysis is successful.

The design of some steel shells, like energy absorbers or bumpers, requires the knowledge of their behavior in large deformations. In this paper, the method of sequential limit analysis is presented and applied to axisymmetric shells in order to study their post-collapse response. Although the material behavior is assumed as rigid-plastic, results compare favorably with those produced by elastic-plastic incremental analyses and the procedure appears to be more efficient and numerically stable. Large displacement effects, both of stable and unstable nature, are implicitly accounted for by mesh updating.

The limit analysis problem, which permits the direct computation of the limit load of rigid-perfectly plastic solids and structures, is considered with reference to anisotropic materials. As a starting step, attention is focused on Hill’s orthotropic condition, for which an explicit expression of the dissipation power in terms of strain rates is established. On this basis, numerical procedures successfully employed in the isotropic (von Mises’) case can be used in this context, as some examples illustrate.

Necessary and sufficient conditions for the occurrence of a bifurcation in the equilibrium path of a discrete structural system are established as a consequence of the degeneracy of the solution of the rate problem at a critical point. Such result is based on the properties of the elastic-plastic rate problem formulated as a linear complementarity problem (LCP) in terms of plastic multipliers (the moduli of the plastic strain rate vectors) as basic unknowns. The conditions here given allow to distinguish, both theoretically and practically, among bounded bifurcations, unbounded bifurcations, limit points, and unloading points. All of the needed quantities depend either on the starting situation or on the actual known term increment; there is no need to compute eigenvalues or eigenvectors of stiffness matrices. The results obtained can be seen as a refinement, for the discrete elastic-plastic problem, of the uniqueness theory given by Hill. The refinement allows covering the case of vector-valued yield fun...

The plane strain problem for elastically isotropic materials with a plastic behavior governed by von Mises condition is examined. By exploiting previous results, a mixed finite element formulation is established in this context. It is shown that if the trial functions for different fields comply with some conditions, the resulting finite element model not only is stable in the Babuska-Brezzi sense, but also retains the same stress redistribution capability of the equivalent displacement model, even if the inclusion relationships for which limitation principles apply are avoided. In particular, it is shown that locking does not occur when, as plasticity spreads, strains become nearly deviatoric and that collapse is correctly predicted in the perfectly plastic case.

In this paper, the results of part I of this study are applied to the 4-node bilinear element. The trial functions for different fields are defined consistently with the a-priori conditions established, the rate problem is generalized to cover finite loading steps and the solution strategy is presented. Computations referred to some discriminating examples from the literature demonstrate that the proposed model is actually capable of predicting the perfectly plastic collapse load for pathological plane strain problems, exhibits excellent performances with coarse meshes and is rather insensitive to mesh distortion.

Si presenta un procedimento per la valutazione numerica del carico di collasso di solidi o strutture costituiti da un materiale schematizzabile come rigido-idealmente plastico. Il metodo si basa sul teorema cinematico del calcolo a rottura, che viene ricondotto alla ricerca del minimo di un funzionale, convesso ma non ovunque differenziabile, sotto vincoli lineari di eguaglianza. La soluzione viene affrontata mediante discretizzazione per elementi finiti. Caratteristiche del metodo proposto sono: (1) il ricorso alla formulazionenaturale dell'approccio agli spostamenti, che consente di cosiderare elementi finiti di ordine comunque elevato; (2) la strategia risolutiva, basata sull'identificazione degli elementi che si mantengono rigidi a collasso, la cui progressiva eliminazione dal problema consente di superare le difficolta connesse con la non regolarita della funzione obiettivo; (3) il ricorso a particolari formulazioni di tipomisto, che eliminano possibili fenomeni dilocking senza compromettere la natura cinematica dell'approccio. Alcuni esempi relativi a problemi piani nelle deformazioni e a piastre di Kirchhoff illustrano le possibilita offerte dal metodo.

Path-dependent materials, complying with Drucker's postulate requirements and governed by an internal variable rate plasticity model, are considered. A variational principle for the small strain, rate plasticity problem is established in this context and extended to cover finite loading steps. Results are subsequently specialized to plane strain solids made of elastically isotropic materials with a plastic behavior governed by the von Mises criterion, accounting for combined isotropic and kinematic hardening. By exploiting previous results, the formulation is fully reduced to the plane. Further generalizations of the statements are also provided, which can be regarded as extensions to the elastic-plastic, plane strain problem of the Hu-Washizu principle in elasticity.

Elastic-plastic (path dependent but time-independent) materials, linear and isotropic in the elastic range and governed by the von Mises yield condition and the relevant associated flow rule, are considered. Both kinematic and isotropic hardening are included. Starting from previous, more general but partial, results and exploiting the purely deviatoric nature of plastic flow, a plane strain version of the constitutive law is formulated in terms of in-plane variables only. The actual plastic strains are replaced by equivalent in-plane measures and the effects of transverse yielding appear as a fictitious, directional kinematic hardening. On this basis, the plane strain elastic-plastic problem can be formulated in a way fully analogous to the equivalent plane stress one: as in the elastic case, differences are confined to the expressions of the constitutive parameters.

This study originated from a specific problem that arose in conjunction with the IRIS (International Reactor Innovative and Secure) project (Carelli et al., 2004; Carelli, 2009). IRIS adopts an integrated primary system reactor (IPSR) configuration with all the primary loop components of a classical Pressurized Water Reactor (PWR) contained inside the vessel (Fig. 1). Among the reactor core internals are the steam generator (SG) units (Cinotti et al., 2002) with the primary fluid flowing outside the tube bundles and subjecting them to significant external pressure. In this situation buckling affects the tube collapse modality and codes become extremely conservative, to the point that up to five years ago design procedures based on the ASME Boiler & Pressure Vessel code (Section III) required an external diameter to thickness ratio (D/t) less than 8.5, leading to an increased thermal resistance in the heat exchange process between primary and secondary fluids, with detrimental consequences on the dimensioning of the heat transfer surface. A reduction in the tube thickness would allow the reduction of the overall heat transfer surface needed to exchange the same amount of power, with consequent saving on tube lengths and/or number of tubes. On the other hand, if the design of the steam generator units is not modified, an increase in the exchanged thermal power and a consequent up rating of the reactor can be obtained. Besides IRIS, other recent proposals for next generation power plants based on PWR technology consider an IPSR design (Ingersoll, 2009; Karahan, 2010; Ninokata, 2006). Such integrated design is particularly suitable for small sized units, i.e., reactors with a power less than 300 MWe following the IAEA’s definition (IAEA, 2007). A significant number of small sized PWR IPSRs is currently under development (e.g., RITM-200, ABV, CAREM, SMART, MRX, NHR-200, Westinghouse SMR, mPower, NuScale, see http://www.world-nuclear.org/info/inf33.html). Packing all the PWR primary components into the reactor pressure vessel (RPV) (Fig. 2) offers several advantages (Ingersoll, 2009): (i) all large coolant pipes are eliminated (only small feed water and steam outlet pipes penetrate the vessel wall); (ii) the total inventory of primary coolant is much larger than for an external loop PWR (this feature increases the heat capacity and thermal inertia of the system and hence yields a much slower response to core heat-up transients); (iii) typically the heat exchangers are placed above the core creating a relatively tall system that facilitates more effective natural circulation of the primary coolant in the case of a coolant pump failure; (iv) the vessel accommodates a relatively large pressurizer volume that provides better control 14

The safety-by-design™ approach adopted for the design of the International Reactor Innovative and Secure (IRIS) resulted in the elimination by design of some of the main accident scenarios classically applicable to Pressurized Water Reactors (PWR) and to the reduction of either consequences or frequency of the remaining classical at-power accident initiators. As a result of such strategy the Core Damage Frequency (CDF) from at-power internal initiating events was reduced to the 10−8 /ry order of magnitude, thus elevating CDF from external events (seismic above all) to an even more significant contributor than what currently experienced in the existing PWR fleet. The same safety-by-design™ approach was then exported from the design of the IRIS reactor and of its safety systems to the design of the IRIS Nuclear Steam Supply System (NSSS) building, with the goal of reducing the impact of seismically induced scenarios. The small footprint of the IRIS NSSS building, which includes all Engineered Safety Features (ESF), all the emergency heat sink and all the required support systems makes the idea of seismic isolation of the entire nuclear island a relatively easy and economically competitive solution. The seismically isolated IRIS NSSS building dramatically reduces the seismic excitation perceived by the reactor vessel, the containment structure and all the main IRIS ESF components, thus virtually eliminating the seismic-induced CDF. This solution is also contributing to the standardization of the IRIS plant, with a single design compatible with a variety of sites covering a wide spectrum of seismic conditions. The conceptual IRIS seismic isolation system is herein presented, along with a selection of the preliminary seismic analyses confirming the drastic reduction of the seismic excitation to the IRIS NSSS building. Along with the adoption of the seismic isolation system, a more refined approach to the computation of the fragility analysis of the components is also being developed, in order to reduce the undue conservatism historically affecting seismic analysis. The new fragility analysis methodology will be particularly focused on the analysis of the isolators themselves, which will now be the limiting components in the evaluation of the overall seismic induced CDF.Copyright © 2009 by ASME

The effect of deck girder slenderness on the lateral buckling load of arched steel bridges is investigated in this article. Models in the literature consider the deck to be infinitely stiff out-of-plane: this reductive hypothesis is removed in this article, and a formulation describing the lateral behaviour of the slender tied arch in terms of two geometric parameters is proposed. A comparison with the relevant Eurocode 3 (EC3) instructions for preliminary stability checks shows that a correction function to include deck slenderness is necessary. Some modifications to code formulations are proposed.

Several engineering applications require cylindrical shells subjected to external pressure, and an increasing need for tubes of significant thickness has been experienced in recent years. So far, little study has been devoted to very stocky tubes, and a great amount of uncertainty exists on some important aspects, such as the consequences of imperfections on their failure pressure. This can only be computed by performing numerical analyses that consider both material (plasticity) and geometric (large displacement) nonlinearities. Such a procedure is feasible, but its use for design purposes is cumbersome, and handy alternatives are worth searching. In this paper, a comparatively simple formula is proposed, based on an interpretation of the relative role that plasticity and instability play in the thick tube range. The formula depends on a crucial coefficient, which can be defined so as to provide safe but reasonably accurate approximations to the collapse pressures computed numerically for tubes made of different metals. The proposal may be useful for preliminary design purposes and can be considered as a first contribution toward a precise assessment of the collapse behavior of tubes in a thickness range so far overlooked.

A class of finite, holonomic elastic-plastic laws is considered and its properties are studied both at the local and at the structural level. This class is fairly general and permits hardening behavior to be covered. A kinematic viewpoint is adopted, resting on geometric compatibility and kinematic admissibility for plastic strains. Two extremum theorems are proved, one characterizing the material response to given strains and the second the structural response to given loads, which can be considered as the kinematic counterpart of static statements already available, at least for the case of perfect plasticity. In particular, the second theorem can be regarded as the dual of the classical Haar-Karman result. Further assumptions which, however, do not appear as unduly restrictive, permit considerable simplications, as two examples illustrate.

A parametric study on the interaction curves for steel beam-columns in fire conditions is performed. The study is based on a trilinear idealization of the uniaxial stress-strain relation for steel at elevated temperature and aims at assessing the influence of the slope of the hardening branch, on which results are strongly dependent. A refined and numerically efficient elastic-plastic beam element model is employed to analyse different load conditions and temperature distributions. Results for uniform temperature are systematic enough to permit fairly general analytical approximations, which might be useful also in the presence of temperature variations along the element length only. Thermal gradients through the cross-section are also considered.

The limit analysis problem, permitting the direct computation of the collapse load of perfectly plastic structures, is considered for anisotropic materials. To this end, the essential piece of information, so far available in a limited number of cases only, is the expression of the dissipation power associated to the anisotropic yield criterion as an explicit function of strain rates. Such expression is derived for conditions based on the Tsai–Wu’s tensor polynomial form for failure criteria, truncated to second order stress terms, but accounting for possible tensile to compressive strength differential. On this basis, numerical procedures successfully employed in the isotropic case can be extended with minor modifications to the anisotropic context, as some examples illustrate.

The numerical study presented in this paper aims at assessing the role played by different aspects on the resistance of bolted joints in shear. To this purpose, a simplified procedure is proposed, based on rigid-plastic limit analysis, used in conjunction with a specifically devised finite element, essentially “plane” but accounting for the effects of transverse strain components. Results show that the model, although resting on drastic simplifying assumptions, permits a better understanding on the influence of a few important aspects affecting the overall joint behavior, such as the role of transverse plastic deformations, the restraining effect of the nut and the contact between the bolt and the metal plate. In particular, it explains how the nut and the washer have a fundamental role in bearing resistance of bolted connections and why pin design requires a different approach.

The paper considers perfectly plastic materials with a yield condition of the form Φ σ =F ij σ ij +F ijkl σ ij σ kl ⩽1 corresponding to a second order truncation of the tensor polynomial expression proposed by Tsai and Wu for failure criteria. Such an expression is often employed for materials exhibiting particular forms of anisotropic failure properties, including orthotropic ones, and accounts for non-symmetric strengths. The limit analysis problem is considered next. The formulation based on the kinematic theorem, reducing to the search of the constrained minimum of a convex functional, was successfully employed in the isotropic case for numerical solutions and can be extended to the present context without modifications, provided that the expression for the dissipation power as an explicit function of strain rates is available. For the material considered, this expression is established in this paper. The result is specialized to plane stress orthotropy and an example is worked out. Although extremely simple, it permits the assessment of the influence of the ratio between tensile and compressive strength and of the inclination of the orthotropy axes with respect to loading directions.

The inherently path-dependent nature of plastic behavior requires that the constitutive laws be formulated in terms of rates. The material response to finite stress or strain changes can only be obtained by integrating the rate equations along a given path. Nevertheless, various elastic-plastic laws in finite terms have been proposed, known as deformation or holonomic theories (among them, those due to Hencky [1] and Nadai [2] are to be mentioned). They must be considered as approximations to the actual path-dependent or non-holonomic behavior, reasonable only along nearly proportional paths.

Conventional displacement finite element formulations for elastic-plastic analysis are faced with the impossibility of enforcing the constitutive relation of rate plasticity locally. This is imposed at a selected set of points only, and, because of significant operational advantages, Gauss integration points are usually employed to this purpose [1]. A common feature of such procedures is that an increase in the number of points where plasticity is enforced, instead of producing more refined models, stiffens the element behavior, sometimes so badly that the model is described as “locking” [2]. Actually, computational experience shows that the best results are obtained by limiting the number of Gauss points to the minimum possible value and that so called reduced integration is preferable in several instances [3].

The basic results of classical (associated) rate plasticity theory are first summarized in this chapter and some variational (minimum) statements are recalled, which provide suitable bases for finite element discretization. Extensions to holonomic or piecewise holonomic representations of the elastic-plastic behavior are next discussed, under the assumption that the constitutive law can be expressed in a piecewise linear form. The discrete elastic-plastic problem for finite load increments is considered in this context and connections with Mathematical Programming formulations are underlined.

A two-field finite element model for the elastic-plastic problem is presented, based on a kinematic minimum theorem due to Maier. The formulation rests on the independent discretization of the displacement and plastic multiplier fields and includes as particular cases existing finite element approaches, which can be conceived as based on implicit particular assumptions for the plastic multiplier model. The implications of such assumptions are discussed and an alternative formulation is presented, able to overcome some inconveniencies which might be experienced with traditional procedures. Mathematical Programming theory can be used to assess some features of the structural elastic-plastic behavior and can be exploited for numerical solution purposes. Some examples illustrate the essential aspects of the behavior of the model proposed and the flexibility of Mathematical Programming formulations.

The impact of computational methods in structural mechanics has been enormously increasing in the last two decades. A new discipline, now labelled as Computational Mechanics, has emerged as a strict intersection between several fields such as theoretical mechanics, computer science, applied mathematics and numerical analysis. In a sense, this discipline has inherited the considerable amount of concepts and knowledge directed toward methods able to provide solutions to problems of technological interest which was produced by researchers in structural mechanics since the last century. Computational mechanics has evolved on this basis (to which Castigliano significantly contributed) to the point of becoming a modern and rapidly developing computer-oriented facet of structural mechanics. The purpose of this paper is to discuss some aspects of this inheritance and to comment on some of the present developments and attitudes in computational mechanics.

In elastic-plastic structures subjected to dynamic external actions, if unbounded plastic deformations are developed, either local failure due to plastic fatigue (alternating plasticity) or gradual divergence of the deformed configuration (incremental collapse) will occur. Therefore, the boundedness in time of total plastic strains, and hence of total plastic work (usually referred to as adaptation or shakedown) is necessary for structural safety, in the sense that it rules out the occurrence of the above critical phenomena. Necessary and sufficient conditions for shakedown have been established by several authors. However, in many instances adaptation is not sufficient to ensure safety. In fact, even if plastic deformations can be proved to be finite, they can exceed some critical limit or exhaust the material ductility. In particular, for dynamic loading histories that cease after a certain time, a structure will certainly shakedown under any load amplitude, so that a safety criterion based on this event is clearly meaningless. Typical histories of this kind are earthquake or blast loadings. When the loading history is known, it is possible, in principle, to assess safety by following the actual plasto-dynamic evolution of the system, but this is often a laborious task and in several cases it provides far more information than is actually needed. On this remark rests the interest of methods capable to provide some essential information, such as upper bounds on maximum deflections or strains, through a moderate computational effort. In recent years several alternative techniques have been developed to bound from the above various quantities of interest. With the exception of very simple situations, the best bounds that can be obtained through any one of these methods involve the solution of constrained optimization problems. In this paper a study of several deformation bounding techniques is performed. The problem is formulated and the main previous results are outlined first with reference to general continua made of hardening materials. Then a class of discrete structural models (such as some finite element discretizations) is considered and, on this basis, two categories of deformation bounding techniques are described from the previous main results. All these techniques, some of which are new, permit the optimization of the upper bound by solving one or more mathematical programming problems of special forms. Some of the bounding procedures are shown to have merely theoretical interest, since they lead to cumbersome numerical procedures or to very coarse bounds. The formulations that appear to have practical application are compared from various standpoints (type of loading history, different hardening rules, influence of second order geometric effects, quantities to be bounded) and first assessment of their practical usefulness is attempted. Generalizations to second-order geometric and thermal effects and to situations in which the time history is not completely known are envisaged.

The paper considers discrete or discretized elastic-plastic structures subjected to conservative external forces and given imposed strains. Material flow laws are assumed to be associated; the plasticity condition and the hardening rule are piecewise linearized.

Two and three dimensional structures are dealt with, subjected to variable repeated loads, in order to establish a numerical tool for determining the load domain multiplier that gives rise to shakedown. The structure is made discrete by finite elements and the yield domain is linearized. By applying Bleich and Melan's theorem, two primal static formulations are found in linear programming, from which the relevant dual kinematic versions are obtained via duality properties. Numerical results are given at the end of the paper, together with some considerations about the numerical efficiency of the proposed formulations.

A finite element model for the elastic–plastic analysis of plane frames is proposed. The formulation is based on the independent modelling of the displacement and plastic strain fields; the latter is modelled both over the cross-section and along the element length as function of a finite number of parameters, which are considered as an extra set of independent variables, in addition to nodal displacements. Stress redistribution is allowed for over the cross-section, but not over the element length, where the distribution of stress resultants (axial forces and bending moments) is imposed consistently with the assumed displacement model; stress redistribution in terms of stress resultants becomes possible only because of the finite number of redundancies introduced when assembling. It is shown that the model can be formulated in such a way that not only compatibility and elasticity, but also equilibrium (in the sense of beam theory), are fully complied with and only the plastic portion of the constitutive relationship is approximately fulfilled, even if, in principle, to any desired level of accuracy. The model produces accurate results, including a detailed representation of the spreading of plastic zones, with a fairly limited number of elements.

The paper is concerned with the application of mathematical programming techniques to a variety of structural plasticity problems. The incremental and finite elastoplastic analysis problems, as well as the plastic limit analysis problem, are formulated as a variety of linear complementarity, quadratic, linear and nonlinear programming problems. A unified computational scheme is presented, and comparisons are made between the mathematical programming techniques and the so-called “direct” methods for nonlinear finite element analysis.

The problem of stress description in displacement finite element models is dealt with. Because of kinematic restraints, such models are able to provide only a global average information on stresses, given in terms of generalized (natural) variables. Expressions for local stress fields as functions of these variables are proposed, which meet a condition of consistency with the assumed displacement model. Such a condition appears as the only requirement which stresses have to comply with and it is shown that it does not define a unique legitimate stress distribution in the element. The implications of possible alternatives are examined with reference to some frequently used plane elements.

Two conceptually different procedures to obtain upper bounds to displacements and deformations developed in the dynamic evolution of elasto-plastic structures are considered. Computationally, both methods require the dynamic analysis of a linear system and a sequence of constrained optimization problems, each roughly analogous to a limit analysis procedure in the quasi-static range, with a substantial saving in effort when compared to an incremental analysis of the plasto-dynamic response. A discrete model for frames is proposed and the two approaches are reformulated in a convenient matrix form. The approaches are then compared on the basis of the information needed (total or plastic displacements, deformations, dissipated energy), of the usefulness of results obtained and of the computational effort involved. The results seem to indicate the practical usefulness of these techniques and encourage the extension of the study to broader classes of structures.

Discrete or discretized structures are considered in the range of large displacements. Elastic plastic behavior is assumed, under the hypothesis that both yield functions and hardening rules are piecewise linear. The structural response to a single finite loading step is assumed to involve regularly progressive yielding (no local unloading). An extremum property of this structural response is established, by recognizing that the relations governing the configuration change coincide with the Kuhn-Tucker conditions of a particular nonlinear constrained optimization problem, subject to sign constraints alone. This extremum property can be regarded as an extension of the theorem of minimum potential energy. Other properties, even if computationally less attractive, broaden the theory developed, so that some results previously obtained are derived as special cases.

Large displacement analysis up to the collapse of trusses is considered. Bars are elastic-plastic and possibly unstable post-critical behaviour is accounted for. The collapse situation may correspond either to a limit point (snap-through buckling) or to the occurrence of bifurcation in Shanley's sense. Attention is mainly focused on the global stability test, which permits the collapse situation to be identified. It is shown that the test can be reduced to a check on the sign of a single number, which can be easily obtained for any given structural configuration. It is also shown how this test can be introduced into a large displacement elastic-plastic analysis code with little additional computational effort.

The configuration of the stinger and ramps of a laying barge can be adjusted by varying the angles between adjacent segments and the arm of each support. On the basis of suitable idealizations and discretizations, a method is developed apt to optimize, for a given laying situation, the choice of the foregoing geometric parameters, alone at given tension or in association with the tensioner thrust. In the former case the function to minimize is the maximum absolute value of the bending moment in the overbend. In the latter case the thrust at the tensioner is minimized for given upper bound on the maximum stress in the overbend. The optimization procedure proposed consists of three phases and allows for unilateral contact between pipe and supports.

Suitable measures of plastic and of total deflections developed over a time interval by an elastic-perfectly plastic continuous structure in the presence of significant inertia and viscous forces, is shown to be bounded from above by a quantity which can be evaluated merely on the basis of a linear-elastic dynamic analysis. The results achieved are extended to elastic-workhardening bodies under the assumption of piecewiselinear yield surfaces. The practical use of the present results is checked comparatively by a simple example.

Constitutive laws for elastic-plastic materials are derived by eliminating the transverse stress component on the basis of the plane-strain constraint. This leads to a fictitious hardening and temperature dependence of the loading function. For standard elastic-plastic materials the resulting laws are associated; however, the plastic strain state is represented by equivalent plastic-strain measures, which also account for transverse yielding. The new constitutive laws, together with the standard reduced form of the equilibrium and compatibility equations, permit the formulation of the plane-strain elastic-plastic analysis problem in terms of the in-plane stress components only. In the case of perfectly plastic materials, the subsequent plane-strain yield surfaces are contained within a domain bounded by a limit surface which represents the yield condition normally adopted in plane-strain limit analysis.

Some implicit or explicit assumptions on which the formulation of compatible finite element models for elastic plastic analysis is based are considered. It is shown that it is possible to formulate global elastic plastic element laws by independently assuming interpolation functions governing the displacement and the plastic strain fields. Some consistency requirements are discussed, which must be enforced in order to avoid artificial stiffening or fictitious redundancy. When these requirements are met, a «consistent» element is produced, whose properties are analyzed. It is also emphasized that, in spite of the use of compatible models, upper bounds to the collapse load are not always obtained for the limit analysis problem. For simplicity, beam elements are considered. However, the formulation holds for continua as well, as discussed in the last section.