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... In practice, since these depend on the previous load history, loading paths used in a nonlinear analysis are somewhat arbitrary in nature. Obtaining the structural state under a particular loading path is therefore rather meaningless [23,24,25]. For this reason, it would be useful a simple theory based on a few data, easily available in engineering practice, and there is nothing more practical than the limit analysis [26,27,28,29,30,31,32]. ...

... The beam nite element adopted (see [25]) uses a stress interpolation ...

... For frames with medium to large span to depth ratios of the members, it is a reasonable approximation [25,6,2,5] to dene the elasto-plastic behavior only in terms of normal stress component σ ≡ σ 11 , while plastic deformations due to shear and torsion are assumed to be negligible. The plastic admissibility condition can be written as ...

This work introduces a mathematical problem named limit fire analysis for estimating the structural safety of 3D frames in case of fire, taking into account the stress redistribution. It is a generalization of the classic limit analysis to a fire event, where the load factor is replaced by the time of fire exposure that reduces the strength of the materials. A lower bound theorem is derived, making it possible to define the limit fire duration, i.e. the maximum time of exposure for which the structure is safe, unaffected by initial plastic deformations, loading history, thermal strains and variations of elastic properties. A numerical framework is also given for evaluating the limit time. The structure is discretized using an equilibrated mixed finite element for each beam and column. The time-dependent admissibility of the stress is imposed through a fiber approach at multiple sections along the elements. An arc-length continuation method with the fire duration as additional unknown provides a sequence of safe time of exposure converging to the limit one. Newton’s iterations are used at each step to determinate the element state and to impose global equilibrium, with all tangent operators obtained analytically. Reinforced concrete frames are considered as example of application. Numerical tests show an efficient analysis also for large buildings.

... A small number of proposals are available to obtain the strength domains of heated sections corresponding to an assigned fire duration or temperature distribution [3,4,5,6]. For well-confined reinforced concrete (RC) sections and steel sections, as proposed in many building standards, the strain limit is sufficiently large to allow an approach based on the classical plasticity theory [7,1,8]. It consists in evaluating a point cloud of generalized yield stresses by assigning the corresponding collapse mechanisms, that is the position and orientation of the neutral axis at the collapse states. ...

... Then, the yield points have to be interpolated in order to handle the yield criteria in structural analysis codes [9,10]. The Minkowski sum [11] of ellipsoids represents an interesting strategy for the approximation of particular convex shapes known as zonoids, such as the cross-section yield surface, as shown in some recent works [1,8]. In this work, we use a new approach for constructing the Minkowski sum. ...

... The material is assumed to be elastic-perfectly plastic with the plastic admissibility condition expressed in terms of normal stress σ 11 only In the following the dependence on s is omitted for a clearer exposition. In accordance with [1,8,7,3], we introduce the plastic mechanism of the cross-section as ...

The starting point of this work is the definition of an automatic procedure for evaluating the axial force-biaxial bending yield surface of steel and reinforced concrete sections in fire. It provides an accurate time-dependent expression of the yield condition by a section analysis carried out once and for all, accounting for the strength reduction of the materials, which is a function of the fire duration. The equilibrium state of 3D frames with such yield conditions, once discretized using beam finite elements, is then formulated as a nonlinear vectorial equation defining a curve in the hyperspace of the discrete variables and the fire duration. An incremental-iterative strategy is proposed for tracing this curve evaluating a sequence of safe states at increasing fire durations up to the limit fire duration, that is the time of exposure which leads to structural collapse. The procedure represents a global fire analysis able to take account of the stress redistribution over the frame. Numerical examples are given to illustrate the proposal.

... However, it is possible, by exploiting the convexity of the yield function, to retain only the convex hull vertexes of the stress envelope. On the basis of this consideration, a computational procedure called Selection Rule Algorithm has been proposed in 30 for identifying the significant vertexes of the convex hull to be considered in a shakedown analysis. ...

... Compared to approaches based on stress resultants, the stress envelope is now easily defined in terms of two scalar values, namely the maximum and minimum effect on the fiber, regardless of dimension and complexity of the load domain and no Selection Rule Algorithm is needed. This key feature guarantees that, as opposite to the existing proposals 30,24,26,25 , the number of load combinations has no influence on the cost of the nonlinear part of the analysis. A decomposition method inspired by the one proposed in 21,22 is applied to solve the global lower bound optimization problem through an incremental-iterative continuation method, similar to a standard incremental elasto-plastic analysis. ...

... The beam finite element adopted (see 30 ) uses a stress interpolation ...

This work presents an efficient fiber analysis for evaluating the shakedown safety factor of three‐dimensional frames under multiple load combinations. Mixed finite elements are employed for an accurate discretization. A continuation method, similar to a standard elasto‐plastic analysis, is used at structural level. It evaluates a pseudo‐equilibrium path made of a sequence of safe states with a converging nondecreasing load factor. Each point of the path is obtained by finding kinematic variables corresponding to self‐equilibrated stresses satisfying Melan's condition for the current load factor to be safe. The stress admissible domain is defined at fiber level as a function of the load factor using the maximum and minimum effect due to all loads. An iterative state determination provides finite element stresses corresponding to assigned kinematic variables and load factor. The overall analysis differs from previous proposals for two novelty points. Firstly, a direct application of the Newton method can be employed, without any need for constrained optimization solvers. Moreover, dimension and complexity of the load domain do not affect the computational cost of the nonlinear analysis. Numerical tests show an accurate estimate of the safety factor using a small number of fibers and an efficient solution also for large buildings.

... For well-confined RC sections, as proposed in many building standards [22,23], the strain limit is sufficiently large to allow an alternative approach based on the classical plasticity theory [24,2,25]. Moreover, it is well known [17,22] that the strain limit gets higher when the temperature increases, making the approach even more appropriate. ...

... The Minkowski sum [30] of ellipsoids represents an interesting alternative for the approximation of particular convex shapes known as zonoids, such as the cross-section yield surface, as shown in some recent works [2,25,24]. In this approach, the generalized stresses are decomposed as a sum of ellipsoidal stress terms and, for each of them, the admissibility condition consists in belonging to the corresponding ellipsoid. ...

... In the following the dependence on s is omitted for a clearer exposition. In accordance with [2,25,24,16], we introduce the plastic mechanism of the cross-section as ...

In this work an automatic procedure for evaluating the axial force-biaxial bending yield surface of reinforced concrete sections in fire is proposed. It provides an accurate time-dependent expression of the yield condition by a section analysis carried out once and for all, accounting for the strength reduction of the materials, which is a function of the fire duration. The equilibrium state of 3D frames with such yield conditions, once discretized using beam finite elements, is formulated as a nonlinear vectorial equation defining a curve in the hyperspace of the discrete variables and the fire duration. A generalized path-following strategy is proposed for tracing this curve and evaluating, if it exists, the limit fire duration, that is the time of exposure which leads to structural collapse. Compared to the previous proposals on the topic, which are limited to local sectional checks, this work is the first to present a global analysis for assessing the fire resistance of 3D frames, providing a time history of the fire event and taking account of the stress redistribution. Numerical examples are given to illustrate and validate the proposal.

... For steel and confined RC section, however, as proposed in many building standards 20,21 , the strain limit is sufficiently large to allow an alternative approach based on the classical plasticity theory 22 . In particular, an accurate approach for the yield surface construction has been recently proposed in 2,23 . It consists in evaluating a point cloud of generalized yield stresses by assigning the corresponding collapse mechanisms, that is position and orientation of the neutral axis. ...

... The Minkowski sum 31 of ellipsoids represents a valid alternative for the approximation of convex shapes, such as the crosssection yield condition, that is for yield surface representable as a sum of segments, as shown in some recent works 2,23,22 . In this approach, the generalized stresses are decomposed as a sum of ellipsoidal stress terms and, for each of them, the admissibility condition consists in belonging to the corresponding ellipsoid. ...

... The beam finite element adopted (see 23 ...

This work makes the Minkowski sum of ellipsoids into a consolidated tool for the representation of the yield surface of arbitrarily shaped composite cross‐sections under axial force and biaxial bending and shows how best to use it within the incremental nonlinear analysis of 3D frames. A geometric interpretation of each term of the sum allows us to construct complex convex surfaces using a low number of ellipsoids, each of them evaluated in a robust and efficient decoupled manner. Specialized algorithms, which exploit the parametrization of the yield surface in terms of cross‐section collapse mechanism, are proposed for an efficient stress update based on the elastic predictor‐return mapping scheme on a 3D beam finite element. The derivation of the algorithmic tangent moduli for the Minkowski sum completes the elements required for an efficient path‐following nonlinear analysis. The proposed methodology is general and can be applied for the representation of any zonoid yield surface and the corresponding implicit stress integration to a variety of structural and plasticity models.

... Moreover, recently developed analysis typologies, involving several load combinations [18] or seismic envelopes [30,31,35], require a great amount of capacity checks; thus, their use in common practice is hampered by the great computational effort of determining a large set of points belonging to the ULS surface. ...

... sections has been formulated in [21]. Originally oriented to finite-element shakedown analysis of 3D framed structures [4], the procedure has been recently enhanced in order to account for Eurocode 8-compliant [10] load combinations [18]. The algorithm reduces the computational effort of the limit analysis by lowering redundant constraints [36,37], without affecting accuracy of the results, and defining the capacity surface by means of a Minkowski sum of ellipsoids. ...

... On the contrary, when used to approximate limit surfaces under the assumption of infinite ductility [18,21], the calibration of support functions becomes straightforward since the surface gradient is known in closed form. ...

The classical Eurocode-compliant ultimate
limit state (ULS) analysis of reinforced concrete
sections is investigated in the paper with the aim of
verifying if and how this well-established design
procedure can be related to plasticity theory. For this
reason, a comparative analysis concerning capacity
surfaces of reinforced concrete cross sections, computed
via a ULS procedure and a limit analysis
approach, is presented. To this end, a preliminary
qualitative discussion outlines modeling assumptions
aiming to reproduce the physical behavior of reinforced
concrete cross sections with respect to ductility
and confinement issues. Besides the theoretical importance
of the proposed approach, numerical experiments
prove that limit analysis yields not only very
accurate results but also a computationally effective
procedure that can be affordably used in common
design practice.

... Use of limit analysis (Malena and Casciaro 2008) was first developed for shakedown analyses of 3D frames (Casciaro and Garcea 2002) and has been enhanced to account for Eurocode 8-compliant load combinations (Leonetti et al. 2015). Such a strategy reduces the computational burden by reducing redundant constraints (Simon and Weichert 2012;Spiliopoulos and Panagiotou 2017) and introduces the possibility of representing the domain boundaries by approximate formulations. ...

Response spectrum analysis represents the preferential strategy to analyze and design civil engineering structures subjected to seismic actions. Nevertheless, most structural codes were developed by following hand computation-oriented philosophies so that their prescriptions can be hard to be implemented in finite element frameworks and often prevent the use of innovative strategies. This contribution presents a review of innovative tools focused on reinforced concrete framed structures aiming to establish a possible organic workflow for design procedures. Some pivotal issues typical of such a structural typology are hereby addressed, and particularly, global torsion and capacity checks in the presence of axial force–biaxial bending responses. This has been done by correlating innovative solutions such as torsional spectra, seismic envelopes, and limit analysis and by presenting a numerical procedure capable of performing capacity checks of reinforced concrete cross sections. The presented strategy aims to be a computationally efficient and exhaustive procedure to be used within the framework of finite element analysis.

... Use of limit analysis (Malena and Casciaro 2008) was first developed for shakedown analyses of 3D frames (Casciaro and Garcea 2002) and has been enhanced to account for Eurocode 8-compliant load combinations (Leonetti et al. 2015). Such a strategy reduces the computational burden by reducing redundant constraints (Simon and Weichert 2012;Spiliopoulos and Panagiotou 2017) and introduces the possibility of representing the domain boundaries by approximate formulations. ...

This book presents a range of research projects focusing on innovative numerical and modeling strategies for the nonlinear analysis of structures and metamaterials. The topics covered concern various analysis approaches based on classical finite element solutions, structural optimization, and analytical solutions in order to present a comprehensive overview of the latest scientific advances. Although based on pioneering research, the contributions are focused on immediate and direct application in practice, providing valuable tools for researchers and practicing professionals alike.

... In this context, limit surfaces are usually computed by algorithms requiring iterative procedures and demanding operations [18,19]. Such an issue is particularly troublesome when the capacity domain must be used in conjunction with recently developed strategies of structural analysis involving a significant amount of capacity checks due to several load combinations [15], seismic actions [25,26,30,32] or iterative procedures [3,10]. Indeed, use of such strategies in common practice is often hampered by the required computational effort. ...

Capacity domains of reinforced concrete elements, computed according to Eurocode 2 provisions, have been investigated from a probabilistic perspective in order to examine the variability of the failure probability over different regions of the domain boundary. To this end, constitutive parameters, such as limit stresses and strains, have been defined as random variables. Discretized ultimate limit state domains have been computed by a fiber-free approach for a set of cross sections. In addition, the failure probability relevant to each point of the domain’s boundary has been evaluated by means of Monte Carlo simulations. The numerical results prove that the failure probability presents a significant variability over the domain boundary; it attains its maximum nearby pure axial force while it drastically decreases in presence of significant bending contributions. Finally, the iso-probability surface, i.e. the locus of the internal forces corresponding to a fixed value of the failure probability, is presented. It permits to establish a probabilistic interpretation of the axial force–biaxial bending capacity check consistently with the underlying philosophy of recent structural codes.

... E' importante poter individuare, in corrispondenza a ciascuna sezione di verifica, le combinazioni potenzialmente più restrittive nell'insieme totale di quelle possibili. Nonè difficile (in linea di principio) se si riflette sulla geometria del problema [4]. ...

Slides del Corso di Formazione "Introduzione all'analisi nonlineare" tenuto presso l'ordine degli Ingegneri di Genova, gennaio 2019

... La stretta corrispondenza fra i due schemi acquista ulteriori implicazioni se ci si riferisce al caso, a cui peraltroè conveniente per altri versi ridursi (vedi [45]), di funzione plastica f [σ] quadratica in σ: ...

Slides del Corso di Formazione "Introduzione all'analisi nonlineare" tenuto presso l'ordine degli Ingegneri di Genova, gennaio 2019

... o, nei casi più complessi, da una somma (alla Minkowski) di ellissoidi D k := D k1 ⊕ D k2 ⊕ · · · . Molto semplicemente (vedi [1,2,3,4,5]), ciò corrisponde a decomporre in parallelo la risposta s k nella somma di più contributi s ki ∈ D ki , ciascuno dei quali soggetto ad una disequazione quadratica: ...

Slides del Corso di Formazione "Introduzione all'analisi nonlineare" tenuto presso l'ordine degli Ingegneri di Genova, gennaio 2019

... , B m in the space of load parameters. It is worth noting that the load domain may not be convex and the detailed descriptions of load domain for complex load conditions can be seen in Ref. [60]. As shown in Fig. 1a, a load domain with five load vertices is taken as an example. ...

In this paper, a novel direct method called the stress compensation method (SCM) is proposed for limit and shakedown analysis of large-scale elastoplastic structures. Without needing to solve the specific mathematical programming problem, the SCM is a two-level iterative procedure based on a sequence of linear elastic finite element solutions where the global stiffness matrix is decomposed only once. In the inner loop, the static admissible residual stress field for shakedown analysis is constructed. In the outer loop, a series of decreasing load multipliers are updated to approach to the shakedown limit multiplier by using an efficient and robust iteration control technique, where the static shakedown theorem is adopted. Three numerical examples up to about 140,000 finite element nodes confirm the applicability and efficiency of this method for two-dimensional and three-dimensional elastoplastic structures, with detailed discussions on the convergence and the accuracy of the proposed algorithm.

... This effective and robust formulation is of particular importance since, when designing structures, there will be cases where the number of independent loads would be much higher than three (e.g. [66,67]). It should also be underlined that the procedure may be embedded in any existing FE code. ...

The Residual Stress Decomposition Method for Shakedown (RSDM-S) is a new iterative direct method to estimate the shakedown load in a 2-dimensional (2D) loading domain. It may be implemented to any existing finite element code, without the need to use a mathematical programming algorithm. An improved and enhanced RSDM-S is proposed herein. A new convergence criterion is presented that makes the procedure almost double as fast. At the same time, the procedure is formulated in a 3-dimensional (3D) polyhedral loading domain, consisting of independently varying mechanical and thermal loads. Using a cyclic loading program that follows the outline of this domain, it is shown that there is hardly any increase in the computational time when passing from a 2D to a 3D domain. Finally, keeping the efficiency, using an alternative cyclic loading program, an automation of the approach to any n-dimensional loading domain is presented. Examples of application are included.

... For instance, a thin-wall continuous beam with softening behavior under one-path loading was analyzed by [25] taking into account material non-linearity and local buckling. Sensitivity analysis of the stability problems of thin-walled structures presented in [17] The right approach is possible either by laborious analyzing of load history in time without any warranty of accounting for the worst histories of independent load cases, or for the entire class of loading as provided in the theory of shakedown analysis (SDA) [1,5,6,8,9,12,[14][15][16]19,20,22,26,27,[29][30][31][32]. The example of such shakedown approach to the steel frames confined with 1st class cross-sections was published in a paper by Atkočiūnas & Venskus [10]; a shakedown limit analysis of the reinforced concrete frames has been done by Alawdin & Bulanov [2]; an updated mathematical model for optimal shakedown analysis of plane reinforced concrete frames according to Eurocodes has been introduced by Alawdin & Liepa [3]. ...

Classical optimization problems of metal structures confined mainly with 1st class cross-sections. But in practice it is common to use the cross-sections of higher classes. In this paper, a new mathematical model for described shakedown optimization problem for metal structures, which elements are designed from 1st to 4th class cross-sections, under variable quasi-static loads is presented. The features of limited plastic redistribution of forces in the structure with thin-walled elements there are taken into account. Authors assume the elastic-plastic flexural buckling in one plane without lateral torsional buckling behavior of members. Design formulae for Methods 1 and 2 for members are analyzed. Structures stiffness constrains are also incorporated in order to satisfy the limit serviceability state requirements. With the help of mathematical programming theory and extreme principles the structure optimization algorithm is developed and justified with the numerical experiment for the metal plane frames.

... In the case of a large-scale system with composite elements these checks may require many iterations and therefore calculating time will significantly increase. The earthquake analysis of buildings and structures can be performed by finding solution of the optimization problem of shakedown analysis, which also takes into account the nonlinear properties of materials [4][5][6][7][8][9][10][11][12][13][14][15]. Such analysis has several advantages: -External actions are introduced as the set of the load cases, that's why we can solve the problem for all direction of seismic load and for every scheme of live load at once; -As part of the solution of the optimization problem we can take into account the elastic-plastic and brittle behavior of the elements [7,15,16]. ...

In this paper the earthquake analysis of composite steel-concrete frames is performed by finding solution of the optimization problem of shakedown analysis, which takes into account the nonlinear properties of materials. The constructions are equipped with systems bearing structures of various elastic-plastic and brittle elements absorbing energy of seismic actions. A mathematical model of this problem is presented on the base of limit analysis theory with partial redistribution of self-stressed internal forces. It is assumed that the load varies randomly within the specified limits. These limits are determined by the possible direction and magnitude of seismic loads. The illustrative example of such analysis of system is introduced. Some attention has been paid to the practical application of the proposed mathematical model.

... He used various codes to find which code gave best results. Leonetti-et-al [13] proposed the efficient treatment of load combination to make the shakedown analysis more affordable tool for practical applications. They used the sap-2000 for analysis. ...

... Structures, during their operational life, are subjected to a sequence of variable actions depicting, sometimes, a very complex loading scenario [2] In this context shakedown analysis furnishes, in a direct way, a reliable safety factor against plastic collapse, loss in functionality due to excessive deformation (ratcheting) or collapse due to low cycle fatigue (plastic shakedown), and also provides valuable information about the internal stress redistribution due to the plastic adaptation phenomenon. . ...

... The earthquake analysis of buildings and structures can be performed by finding solution of the optimization problem of shakedown analysis, which also takes into account the nonlinear properties of materials (Aliawdin 2005, Atkociunas 2011, Cyras 1983, Fadaee, M. J., et al. 2008, König 1987, Leonetti et al. 2015, Nguyen 2006, Tangaramvong et al. 2013, Weichert and Ponter 2009. Such analysis has several advantages: ...

Shakedown analysis plays a substantial role in safety assessment, especially in nuclear plant industry, chemical industry and civil engineering. This paper presents numerical investigations of shakedown problems with holes using the adaptive extended isogeometric analysis (XIGA). An adaptive strategy performed in the discretized framework of a kinematic approach and second-order cone programming (SOCP) is presented. The main idea is to integrate XIGA based on local refined non-uniform rational B-splines (LR NURBS) into the SOCP to form an efficient approach for shakedown analysis. A SOCP form based on the von Mises yield criterion is established. By means of an effective SOCP solver, the arising optimization problem is then solved. In addition, the level set function is adopted to detect the position of holes. The L2-norm of plastic strain rates is used to guide local mesh refinement. The merits of the developed approach are its easy implementation and high computational efficiency. We consider several benchmark examples to illustrate the accuracy, reliability and efficiency of the developed method.

This book provides an overview of direct methods such as limit and shakedown analysis, which are intended to do away with the need for cumbersome step-by-step calculations and determine the loading limits of mechanical structures under monotone, cyclic or variable loading with unknown loading history. The respective contributions demonstrate how tremendous advances in numerical methods, especially in optimization, have contributed to the success of direct methods and their practical applicability to engineering problems in structural mechanics, pavement and general soil mechanics, as well as the design of composite materials. The content reflects the outcomes of the workshop “Direct Methods: Methodological Progress and Engineering Applications,” which was offered as a mini-symposium of PCM-CMM 2019, held in Cracow, Poland in September 2019.

Seconda parte delle slides del corso di approfondimento tenuto presso l'Ordine degli Ingegneri di Cosenza nei giorni 27 e 28 febbraio 2020

Prima parte delle slides del corso di approfondimento tenuto all'Ordine degli Ingegneri di Cosenza nei giorni 27 e 28 febbraio 2020

Testo della presentazione tenuta in Roma presso l’Ordine degli Ingegneri della Provincia

Up to date, classical optimization problems of metal structures confined mainly with 1st class cross-sections. But in practice it is common to use the cross-sections of higher classes. In this paper, a new mathematical model for described shakedown optimization problem for metal structures, which elements are designed from 1st to 4th class cross-sections, under variable quasi-static loads is presented. The features of limited plastic redistribution of forces in the structure with thin-walled elements there are taken into account. Authors assume the elastic-plastic flexural buckling in one plane without lateral torsional buckling behavior of members. Design formulae for Methods 1 and 2 for members are analyzed. Structures stiffness constrains are also incorporated in order to satisfy the limit serviceability state requirements. With the help of mathematical programming theory and extreme principles the structure optimization algorithm is developed and justified with the numerical experiment for the metal plane frames.

Abstract The present contribution advocates an up-scaling procedure for computing the limit loads of spatial structures made of composite beams. The resolution of an auxiliary yield design problem leads to the determination of a yield surface in the space of axial force and bending moments. A general method for approximating the numerically computed yield surface by a sum of several ellipsoids is developed. The so-obtained analytical expression of the criterion is then incorporated in the yield design calculations of the whole structure, using second-order cone programming techniques. An illustrative application to a complex spatial frame structure is presented.

In our previous paper the implicit corotational method (ICM) was presented as a general procedure for recovering objective nonlinear models fully reusing the information obtained by the corresponding linear theories. The present work deals with the implementation of the ICM as a numerical tool for the finite element analysis of nonlinear structures using either a path-following or an asymptotic approach. Different aspects of the FEM modeling are discussed in detail, including the numerical handling of finite rotations, interpolation strategies, and equation formats. Two mixed finite elements are presented, suitable for nonlinear analysis: a three-dimensional beam element, based on interpolation of both the kinematic and static fields, and a rotation-free thin-plate element, based on a biquadratic spline interpolation of the displacement and piece-wise constant interpolation of stress. Both are frame invariant and free from nonlinear locking. A numerical investigation has been performed, comparing beam and plate solutions in the case of thin-walled beams. The good agreement between the recovered results and the available theoretical solutions and/or numerical benchmarks clearly shows the correctness and robustness of the proposed approach as a general strategy for numerical implementations.

The paper describes the application of the Linear Matching Method to the direct evaluation of limits associated with an elastic-perfectly plastic body subjected to cyclic loading. Methods for limit load and shakedown limit are followed by ratchet limits. The method is distinguished from other programming methods by ensuring that equilibrium and compatibility are satisfied at each stage. The method has been extended beyond the range of most other direct methods by including ratchet limits and high temperature material behaviour. Implementation is possible within the user routines of commercial finite element codes. The paper emphasise the theoretical characteristics of the method and discusses significant aspects of convergence, both theoretical and numerical. The application of the method to industrial Life Assessment problems and to geotechnical problems is summarized.

The load bearing capacity of engineering structures subjected to varying thermo-mechanical loading can be determined most conveniently by shakedown analysis. Admittedly, if limited kinematical hardening is considered, shakedown analysis is, as yet, restricted to either one or two independently varying loads. In consequence, the aim of this paper is to extend existing formulations for arbitrary numbers of loading, which is inevitable for most technical applications. For this, limited kinematical hardening was incorporated into Melan’s shakedown theorem by means of a two-surface model covering both alternating plasticity and ratcheting, generalized to n-dimensional loading spaces. As a result, the three-dimensional shakedown domain accounting for hardening is presented for a flanged pipe subjected to thermo-mechanical loading.

A new geometrically nonlinear model for homogeneous and isotropic beams with generic section including non-uniform warping due to torsion and shear is derived. Each section is endowed with a corotational frame where statics and kinematics are described using a 3D linear elastic model which extends the Saint-Venant solution to non-uniform warping cases. The algebra of change of observer and a mixed variational principle give the model in terms of generalized parameters. Using a mixed interpolation the model is implemented within a FEM Koiter analysis highly sensitive to the geometrical coherence of the formulation.

A model for beams with heterogeneous and anisotropic cross-section is presented. A semi-analytical approach which exploits a FEM discretization of the cross-section is used to derive a Ritz–Galerkin approximation of the stress and displacement fields. In this way the Saint-Venant solution is easily generalized to generic composite sections and additional variable warping effects are also included in the model. Numerical results relative to composite beams with compact or thin walled section are presented and compared with full 3D analyses.

A linear model for beams with compact or thin-walled sections and heterogeneous anisotropic materials is presented. It is obtained by means of a Ritz–Galerkin approximation using independent descriptions of the stress and displacement fields. These are evaluated by a preliminary semi-analytic solution based on a finite element description of the cross section. A coherent definition of the deformations and stresses is obtained which includes both the generalized Saint Venànt solution for generic materials and some significant additional effects, due to out-of-plane warping and section distortions. The so-built 1-D model maintains the richness of the 3-D solution using a small number of variables.

Determining the load-bearing capacity of engineering structures is essential for their design. In the case of varying thermo-mechanical loading beyond the elastic limit, the statical shakedown analysis constitutes a particularly suitable tool for this. The application of the statical shakedown theorem, however, leads to a nonlinear convex optimization problem, which is typically characterized by large numbers of variables and constraints. In the present work, this optimization problem is solved by a primal–dual interior-point algorithm, which shows a remarkable performance due to its problem-tailored formulation. Nevertheless, the solution procedure remains still a demanding task from computational point of view. Thus, the aim of this paper is to tackle the task of solving large-scale problems by use of a new so-called selective algorithm. This algorithm detects automatically the plastically most affected zones within the structure, which have the highest influence on the solution. The entire system is then reduced to a substructure consisting of these zones, based upon which a new optimization problem can be set up, which is solved with significantly less effort. Consequently, the running time decreases drastically, as is shown by application to numerical examples from the field of power plant engineering.

A numerical method for the computation of shakedown loads of structures subjected to varying thermal and mechanical loading is proposed for the case of multidimensional loading spaces. The shakedown loading factors are determined based on the lower bound direct method using the von Mises yield criterion. The resulting nonlinear convex optimization problem is solved by use of the interior-point method. Although the underlying theory allows for the consideration of arbitrary numbers of loadings in principle, until now applications have been restricted to special cases, where either one or two loads vary independently. In this article, former formulations of the optimization problem are generalized for the case of arbitrary numbers of loadings. The method is implemented into an interior-point algorithm specially designed for this method. For illustration, numerical results are presented for a three-dimensional loading space applied to a square plate with a central circular hole.

The asymptotic steady state behavior of an elastic–perfectly plastic structure under cyclic loading may be determined by time consuming incremental time-stepping calculations. Direct methods, alternatively, have a big computational advantage as they attempt to find the characteristics of the cyclic state right from the start of the calculations. Most of these methods address an elastic shakedown state through the shakedown theorems and on the basis of mathematical programming algorithms. In the present paper, a novel direct method that has a more physical basis and may predict any cyclic stress state of a structure under a given loading is presented. The method exploits the cyclic nature of the expected residual stress distribution at the steady cycle. Thus, after equilibrating the elastic part of the total stress with the external load, the unknown residual stress part is decomposed into Fourier series whose coefficients are evaluated iteratively by satisfying compatibility and equilibrium with zero loads at time points inside the cycle and then integrating over the cycle. A computationally simple way to account for plasticity is proposed. The procedure converges uniformly to the true cyclic residual stress for a loading below the elastic shakedown limit or to an unsafe cyclic total stress, which may be used to mark the regions with plastic straining inside the cycle. The method then continues to determine whether the applied loading would lead the structure to ratcheting or to regions that alternate plastically. The procedure is formulated within the finite element method. A von Mises yield surface is typically used. Examples of application of one and two dimensional structures are included.

We present a numerical method for the computation of shakedown loads of engineering structures with limited kinematical hardening under thermo-mechanical loading. The method is based on Melan’s statical shakedown theorem, which results in a nonlinear convex optimization problem. This is solved by an interior-point algorithm recently developed by the authors, specially designed for lower bound shakedown analysis of large-scale problems. Limited kinematical hardening is taken into account by use of a two-surface model, such that both alternating plasticity and incremental collapse can be captured. For the yield surface as well as for the bounding surface the von Mises criterion is used. The proposed method is validated by two examples, where numerical results are compared to those of literature where available.

The finite element method discretized static shakedown analysis of steel constructions leads to large, sparse convex optimization problems. Under the von Mises yield criterion, they lead to second-order cone programming problems, for which the most appropriate techniques are Interior Point Methods. Various approaches exploiting the specific characteristics of the shakedown problems are presented and discussed.

In the present paper, the method proposed in [Casciaro R, Garcea G. An iterative method for shakedown analysis, Comput Methods Appl Mech Eng 2002;191:5761–92] for the shakedown analysis is used for the evaluation of the shakedown safety factor of reinforced concrete 3D frames subjected to a combination of varying loads. The FEM discretization for 3D frames is obtained subdividing the frame into rods and using a 3D beam finite element. Beam end-sections are used for checking plastic admissibility and for this reason the formulation models the interaction between the axial force and biaxial bending by a piecewise linearization of the elastic domain, for generic shape reinforced concrete sections. Concrete and steel are assumed to be elasto-plastic, and, for concrete, tensile resistance is ignored. The algorithm is described in detail and some numerical results, that show the efficacy and effectiveness of the proposed method, are reported and discussed.

The paper describes a primal–dual algorithm for shakedown analysis of structures with the use of kinematically admissible finite elements. Starting from Koiter kinematic theorem, a primal–dual optimization condition is established by adopting static terms as complementary variables. Newton-like iterations are performed to give both upper and quasi-lower bounds of the load factor that converge rapidly to the accurate solution of shakedown limit.

This paper is devoted to propose an algorithm to solve the discrete form of the shakedown analysis problem with a nonlinear yield function. Firstly, variational formulations for shakedown analysis of structures under variable loads are considered. Secondly, the corresponding discrete forms are briefly recalled. Then, the algorithm is derived by combining a Newton formula based on the discrete equality conditions with a return mapping procedure focusing plastic admissibility.The proposed numerical procedure can be applied to finite element models with large number of degrees of freedom because the algorithm is specially designed to take advantage of the structure of such models. The numerical procedure is applied to a square plate with a circular hole under variable traction in two directions, and the obtained approximations are compared with available results. Finally, a rectangular plate with two small holes is considered in both plane strain and plane stress conditions. All application use the Mises condition without linearization.

In a companion paper, a response-spectrum-based procedure for predicting the envelope that bounds the simultaneous action of two or more seismic responses was developed. in the present paper, we evaluate the accuracy of the procedure and investigate its influence on the design of structural elements subjected to seismic loads. The accuracy of the proposed envelope is examined by means of comparison with time-history analyses using simulated and recorded ground motions. It is found that the proposed envelope has a level of accuracy that is commensurate with its response spectrum bases. The significance of the proposed envelope is demonstrated by designing the columns of an example reinforced concrete structure by the proposed and conventional methods. In particular, the required reinforcement ratio for each column is determined by superimposing the response envelope that bounds the axial force and biaxial bending moments acting in the column onto a prescribed moment-axial capacity surface. Savings as high as 50% in the required reinforcement ratio are gained by use of the proposed method.

In the design or analysis of structures for seismic loads, the effects of forces acting simultaneously in a member must be considered. The most common example is in the interaction of bending moments and axial load in columns. The usual response spectrum method provides the maximum values of individual responses, but the critical combination of these responses may not involve any of these maxima. In this paper, a response-spectrum-based procedure for predicting the envelope that bounds two or more responses in a linear structure is developed. It is shown that, for an assumed orientation of the principal axes along which the ground motion components are uncorrelated, this envelope is an ellipsoid. For the case when the orientation of the principal axes is unknown, a "supreme" envelope is derived, which corresponds to the most critical orientation of the axes. The response envelope can be superimposed on a capacity curve to determine the adequacy of a given design. In a companion paper, the results of a numerical study are presented to illustrate the accuracy and significance of this method.

This work concerns finite element limit and shakedown analysis of spatial steel frames under nonlinear yield criteria of steel sections. Inner and outer ellipsoidal approximations to the plastic interaction yield surfaces are systematically constructed. Under ellipsoidal yield criteria, the arising computational optimization problem becomes a second-order cone programming problem, for which free and commercial software packages are available, capable of treating large-scale problems. The present study is focused on approximations to the nonlinear plastic interaction provisions of Eurocode 3. Moreover, a well-known criterion due to Orbison is considered. Examples of limit and shakedown analysis of spatial frames under the aforementioned ellipsoidal approximations are presented and several aspects are discussed. For comparison, a fairly general interior point algorithm is also successfully applied to the limit and shakedown analysis under the original, non-ellipsoidal form of the Orbison criterion.

A new linear model for beams with compact or thin-walled section is presented. The formulation is based on the Hellinger–Reissner principle with independent descriptions of the stress and displacement fields. The kinematics is constituted by a rigid section motion and non uniform out-of-plane warpings related to shear and torsion. The stress field is built on the basis of the Saint-Venànt (SV) solution and with a new part to describe the variable warping.The formulation of a finite element with exact shape functions made possible to validate the beam model avoiding discretization errors.

We present the preliminary results of a novel approach to the state determination of polygonal sections of arbitrary shape endowed with elasto-plastic uniaxial constitutive laws.
By means of a suitable application of Gauss theorem, we prove that the normal stress resultants can be computed analytically as sum of finite quantities evaluated solely at the vertices of the section. For this reason, the proposed approach has been termed fiber-free to emphasize the fact that it does not require any subdivision of the section in fibers.
Numerical results show that the fiber approach is grossly inaccurate and that the number of fibers required to achieve a degree of accuracy comparable with that entailed by the fiber-free approach is at least one order of magnitude greater than the one commonly suggested in commercial software for nonlinear frame analysis. Copyright

A three field variational framework is used to formulate a new class of mixed finite elements for the elastoplastic analysis of 2D problems. The proposed finite elements are based on the independent interpolation of the stress, displacement and plastic multiplier fields. In particular new and richer interpolating patterns are proposed for the plastic multiplier in order to go beyond the oversimplifying constant assumption used in Bilotta and Casciaro (2007) [1]. As will be shown, this last choice is useful to simplify the return mapping algorithm but adversely affect the accuracy of the finite element. More articulated interpolations transform the return mapping process into a convex nonlinear optimization problem with few variables and constraints, a problem that can be efficiently solved using optimization algorithms, without penalizing overall computational efficiency. Several kinds of interpolations are proposed and compared with respect to accuracy and efficiency by performing a series of numerical tests on plane stress/strain problems modeled on the basis of the von Mises and Drucker–Prager yield functions. The reliability and good performance of the proposed elements are evident.

Finite element asymptotic post-buckling analysis, being based on fourth-order expansions of the strain energy, requires that nonlinear structural modeling be accurate to same order, at least with respect to the rigid motions of the elements. A corotational description is proposed here as a general tool to satisfy this requirement of objectivity, by referring each element to a local frame which moves (rotates) with the element, so filtering its rigid motion. In this description nonlinearity of the problem derives essentially from the change of reference, from the global fixed frame to the local one, the strain energy being governed by their relative rotations. In finite kinematics, this noticeably complicates the algebra for obtaining exact expressions of its variations.Quite simple, basic expressions for the first four corotational derivatives of the strain energy are provided, allowing the set up of a fourth-order accurate asymptotic analysis starting from standard finite elements based on linear or simplified nonlinear local modelings. The formulation is implemented for the analysis of 3D beam assemblages and several numerical results are presented and discussed showing the effectiveness and robustness of the proposed approach in reproducing the nonlinear equilibrium path in both cases of monomodal and coupled multimodal buckling.

A numerical approach for limit analysis of structures whose constituent material exhibits orthotropic behaviour is presented and discussed. Attention is focused on orthotropic composite laminates under plane stress conditions. The proposed approach is an extension, in the context of orthotropic materials, of the linear matching method (LMM). The latter is based on a sequence of linear analyses performed on the analysed structure made of a fictitious linear viscous material with spatially varying moduli. Here the LMM is applied to structures made of materials obeying the Tsai–Wu criterion. An appropriate choice of the fictitious material, which in this case is assumed linear, viscous, orthotropic and suffering a distribution of assigned initial stresses, reduces the number of parameters to be spatially varied thus rendering the whole procedure applicable and reliable. The results obtained are highly promising as witnessed by a number of numerical examples which are carried out to verify the effectiveness of the proposed approach. Copyright © 2006 John Wiley & Sons, Ltd.

A mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in an FEM context is presented. From the optimization point of view, standard arc-length strain-driven elastoplastic analyses, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other non-linear programming methods, simpler convergence proofs and duality to be exploited. Owing to the unified approach in terms of total stresses, the strain-driven algorithms become more effective and less non-linear with respect to a self-equilibrated stress formulation and easier to implement in the existing codes performing elastoplastic analysis. The elastic domain is represented avoiding any linearization of the yield function so improving both the accuracy and the performance. Better results are obtained using two different finite elements, one with a good behavior in the elastic range and the other suitable for performing elastoplastic analysis. The proposed formulation is compared with a specialized implementation of the primal–dual interior point method suitable to solve the problems at hand. Copyright © 2011 John Wiley & Sons, Ltd.

In the present paper, the formulation proposed by Casciaro and Garcea (Comput. Meth. Appl. Mech. Eng., 2002; 191:5761–5792) and applied to the shakedown analysis of plane frames, is extended to the analysis of two-dimensional flat structures in both the cases of plane-stress and plane-strain. The discrete formulation is obtained using a mixed finite element in which both stress and displacement fields are interpolated. The material is assumed to be elasto-plastic and a linearization of the elastic domain is performed. The result is a versatile iterative scheme well suited to implementation in general purpose FEM codes. An extensive series of numerical tests is presented showing the reliability of the proposed formulation. Copyright © 2005 John Wiley & Sons, Ltd.

Based on the lower-bound shakedown theorem by Melan, a method to analyse pavements under cyclic, in particular, rolling contact loading is presented. Repeated sliding/rolling line contact as well as repeated stationary contact is considered. The material is assumed to be rate-independent elastic–plastic. As yield conditions, the rounded Mohr–Coulomb and von Mises yield criteria are used, assuming associated flow rules. The proposed numerical method is based on finite elements, and the inherent optimization problem to determine the shakedown factors is solved using the interior-point method. Several numerical results are presented and compared with the existing results in literatures. Copyright © 2008 John Wiley & Sons, Ltd.

Shakedown analysis for elastic–perfect plastic structures is discussed and a fast incremental-iterative solution method is proposed, suitable for the FEM analyses of large structures.The theoretical motivations of the proposed method are discussed in detail and an example of its implementation is described with reference to plane frame analysis.Some numerical results are presented showing the numerical performances of the method.

A generalization of Melan’s shakedown theorem is presented taking into account geometrical effects and plastic ductile damage. Numerical results illustrate the proposed method.

We present a family of algorithms for evaluating the ultimate limit state of composite and reinforced concrete (RC) sections of arbitrary polygonal or circular shape, either simple or multicell, subject to axial force and biaxial bending. All algorithms are based upon a secant strategy for computing the stiffness matrix used in the equilibrium iterations while they differ in the scheme adopted to estimate the ultimate values of axial force and bending moments. In this respect they enhance in several ways two analogous algorithms presented in a previous paper [L. De Vivo, L. Rosati, Ultimate strength analysis of reinforced concrete sections subject to axial force and biaxial bending, Comput. Methods Appl. Mech. Engrg. 166 (1998) 261–287] for reinforced concrete sections. First, it is shown the specialization of some new formulas, proved elsewhere, for evaluating the entries of the secant stiffness matrix solely as function of the position vectors of the vertices of the section, assumed to be polygonal, and of the constitutive parameters of the nonlinear stress–strain laws for the materials. Second, the case of sections made of or containing either circles or segments of circular arc is exactly taken into account without the need of approximating such shapes by polygons. Third, the extensive numerical tests carried out for a wide range of RC sections and for a benchmark steel–concrete section have shown that the convergence rate and the stability of the new algorithms considerably increase with respect to the ones presented in De Vivo and Rosati (1998). As outcome of the numerical experiments, two solution strategies have been selected as the optimal ones since they combine both global convergence and a satisfactory convergence rate.

An arch is often connected with other structural members and may be supported by elastic foundations that provide elastic-like restraints to the arch. These elastic restraints participate in the structural response of the arch, and may significantly influence its structural behaviour. A curved-beam element for the structural analysis of arches should therefore consider the effects of these elastic restraints. It is known that the elasto-plastic properties of materials also affect the behaviour of a structure significantly, and so the nonlinearities of the materials need to be included in the curved element for elasto-plastic analysis of an arch. An elastic curved-beam element for the in-plane nonlinear analysis of arches is extended here to an elasto-plastic curved-beam element by formulating the elastic restraints and the material nonlinearities into the element. Geometric nonlinearities in the element are based on an accurate rotation matrix of the two dimensional special orthogonal group that satisfies the desired orthogonality and unimodular conditions. Comparisons of results from the curved element with test results and with analytical solutions demonstrate that the curved-beam element can provide accurate predictions for elastic behaviour, and acceptablly accurate predictions for the elasto-plastic behaviour of elastically restrained arches.

The paper deals with the use of Saint Venànt’s general rod theory for deriving the stiffness matrix for 3D beam elements with general cross-section. The elastic factors of the section are obtained through the numerical solution of the Saint Venànt differential equations. Different discretization strategies have been investigated including FEM and BEM alternatives the one based on 6-node triangular elements appears to be the best choice.

Encyclopedia of computational mechanics

- N Zouain

Zouain N. Encyclopedia of computational mechanics. John Wiley & Sons; 2004.
p. 291-334, [chpter Shakedown and safety assessment].

EN 1991 – Eurocode 1: actions on structures

- European Union

European Union. EN 1991 – Eurocode 1: actions on structures; 1991.