Advances in Direct Methods for Limit States of Structures and Materials
Algorithms and Applications
- 2026
- Book
- Editors
- Konstantinos V. Spiliopoulos
- Dieter Weichert
- Publisher
- Springer Nature Switzerland
About this book
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 contents reflect the outcomes of the 8th Workshop “Direct Methods in Limit States of Structures and Materials,” held in Athens, Greece in September 2024.
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Table of Contents
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Frontmatter
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On Shakedown of Shape Memory Alloys Structures with Functional Fatigue—Application to Nitinol Stents
Michaël PeigneyThis chapter delves into the complex behavior of shape memory alloys (SMAs), focusing on their response to cyclic loading conditions. The study introduces static and kinematic shakedown theorems that account for both structural and functional fatigue, providing a comprehensive framework for analyzing the long-term performance of SMA structures. The research highlights the unique properties of SMAs, such as the shape memory effect and superelastic behavior, and explores how these properties are influenced by phase transformations between austenite and martensite. The chapter also discusses the application of these theorems to nitinol stents, a critical component in biomedical engineering. The study concludes with a detailed analysis of the shakedown domain for nitinol stents under mixed pressure-bending loading, offering valuable insights into the fatigue life of these devices. The research provides a novel approach to understanding the behavior of SMAs under cyclic loading, making it a crucial resource for professionals in materials science and engineering.AI Generated
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AbstractShape memory alloys (SMAs) present exciting opportunities across a variety of fields, including aeronautics, robotics, biomedical engineering, and structural engineering. The unique properties of these materials arise from a solid-to-solid phase transformation that occurs at the microscopic level. Modeling the phase transformation in SMAs is a complex topic. Recently, SMA models that couple phase transformation with permanent inelasticity have been developed to account for degradation effects commonly observed during cyclic loading—a phenomenon known as functional fatigue. In this paper, we present some extensions of the classical static and kinematic shakedown theorems of plasticity to such material models. These extended results provide conditions under which energy dissipation remains bounded, offering significant benefits for the fatigue life of SMAs. As an application, we investigate the shakedown behavior of a nitinol stent subjected to combined pressure-bending loads. The study demonstrates how the proposed approach can be integrated with finite-element analysis to examine the shakedown behavior of complex three-dimensional structures, offering practical insights into the design and durability of SMA-based systems. -
A Crystal Plasticity Based Lower Bound Direct Method and Its Application in the Fatigue Strength Prediction of an Aluminum Alloy Material
Shengzhen Xin, Lele Zhang, Geng ChenThis chapter introduces a novel direct method for crystal plasticity based shakedown analysis, integrating the lower bound theorem with a rate-independent crystal plasticity model. The method is designed to predict the fatigue strength of polycrystalline materials by considering dislocation slip as the yield criterion. The effectiveness of the method is verified using traditional incremental simulation, confirming that the results correspond to the elastic shakedown limit. The chapter also explores the localization of cyclic elastoplastic behavior in polycrystalline materials and the size effect of statistically equivalent representative volume elements (SERVEs) on the predicted shakedown limits. The method is applied to predict the shakedown limits of AlSi10Mg materials fabricated via Laser Metal Deposition (LMD), demonstrating its practical application and robustness. The study concludes that the method can be employed to evaluate other face-centered cubic (FCC) polycrystalline alloys and materials with body-centered cubic (BCC) or hexagonal close-packed (HCP) structures, as long as the strain-rate-dependence is insignificant.AI Generated
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AbstractStrength prediction of materials is an important application of shakedown analysis. Inspired by Dang Van's theory, numerous studies have evaluated endurance limits of materials by direct methods (DMs) based on the Von-Mises or Drucker-Prager yield criterion. It is worth noting that very few researches could implement the Dang Van’s view for shakedown analysis of polycrystalline materials using DMs from the perspective of crystal plasticity (CP). To this end, in the present paper we developed a direct method (CP-DM) for crystal plasticity based shakedown analysis. In this approach, the lower bound theorem is integrated with a crystal plasticity constitutive model, and the CP-DM is constructed based on the strain rate independence and the kinematic hardening of materials. Taking the AlSi10Mg made by laser melting deposition (LMD) as the exemplary material, statistically equivalent representative volume elements (SERVEs) of polycrystals were generated, and crystal plasticity parameters were calibrated by experimental tensile curves. In several case studies, the proposed numerical method was carefully verified by incremental analysis, and the rate-independence of the constitutive model was robustly checked. The verified method was employed to statistically predict the shakedown limits of LMDed AlSi10Mg. This paper confirmed the statistical robustness of the derived results through analyzing the size effect of SERVEs. The numerical method developed in this study can be a promising approach to directly predict shakedown limits of rate-independent polycrystalline materials. -
Limit Analysis of 2D Problems Using a Hybrid Virtual Element Formulation
Francesco S. Liguori, Antonio Madeo, Sonia Marfia, Giovanni Garcea, Elio SaccoThis chapter delves into the Virtual Element Method (VEM) and its application in solving 2D problems in mechanics. It highlights the method's ability to use boundary variables and its tolerance for high distortion, making it suitable for various innovative applications. The text introduces the Hybrid Virtual Element Method (HVEM), a stabilization-free formulation that combines the advantages of VEM with the robustness of mixed formulations. The HVEM is applied to 2D plasticity problems, and its accuracy and robustness are demonstrated through numerical results. The chapter also compares HVEM with classical VEM formulations, showcasing its superior performance and absence of locking phenomena. The convergence analysis reveals a high degree of accuracy, even with coarse meshes, making HVEM a promising approach for limit analysis in mechanics.AI Generated
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AbstractThis work presents a limit-analysis formulation for the Hybrid Virtual Element Method (HVEM). The key features of the approach are the use of an energy norm for the VE projection and a high-order, divergence-free stress interpolation. Unlike classical VEM formulations, HVEM eliminates the need for stabilization terms, thereby avoiding issues related to the choice of stabilization parameters. The method effectively avoids volumetric locking and spurious hardening effects observed in stabilized VEM approaches. The self-equilibrated assumed stress field is suitable for a limit analysis based on the static theorem. Plastic admissibility is tested in a sufficiently large number of control points inside the element domain. The limit-analysis problem is solved through a proximal point formulation, allowing it to be rewritten in a pseudo elasto-plastic form. Numerical experiments demonstrate the accuracy of HVEM even on coarse meshes and its high convergence rate in predicting collapse loads. -
Evaluation of the Torsional Strength of Thin-Walled RC Beams in Fire Conditions
Sabine Boulvard, Duc Toan Pham, Jérémy BleyerThis chapter delves into the evaluation of torsional strength in thin-walled reinforced concrete (RC) beams under fire conditions, a critical yet often overlooked aspect of structural design. The text begins by highlighting the potential for torsional failure in RC structures during fires, despite the rarity of such events, and the inadequacy of current design methods. It then explores the link between shear strength and torsional strength, proposing a method based on yield design theory for the torsional design of thin-walled RC beams under fire conditions. The chapter presents two finite-element models and a semi-analytical formulation for calculating torsional strength, comparing their results with those of existing methods. It also discusses the influence of fire exposure on torsional strength and the validity of the proposed method. The text concludes by highlighting the potential of the proposed method for improving the torsional design of thin-walled RC beams under fire conditions, while acknowledging the need for further experimental validation and parametric studies.AI Generated
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AbstractIn this contribution, a semi-analytical solution and two numerical models are applied to a thin-walled reinforced concrete (RC) beam in order to estimate its torsional moment strength in fire conditions. The semi-analytical solution and one of the numerical models rely on computing the shear strength of the four walls of the beam separately, while the second numerical model is applied to the whole cross-section considering the walls joined at the corners. The three methods are compared to each other, before being compared with an experimental result found in the literature. The comparison with the experimental result is given for several values of the effectiveness factor for the compressive strength of concrete in order to assess the relevance of the finite-element models and, indirectly, the semi-analytical solution for the torsional design of thin-walled RC beams. -
A Non-incremental Numerical Method for Non-associated Elastoplastic Problems Using the SBEN Principle and the Bipotential
Danial Hussain, Abdelbacet Oueslati, Pierre Gosselet, Géry de Saxcé, Djimedo KondoThis chapter introduces a non-incremental numerical method for solving non-associated elastoplastic problems using the SBEN principle and bipotential. The method addresses challenges faced by traditional step-by-step approaches, such as convergence issues and high computational costs. The text begins with an overview of symplectic geometry and the concept of bipotential, followed by the symplectic BEN variational principle. It then applies this principle to non-associated plasticity in quasi-static problems, focusing on a thick-walled tube under internal pressure. The chapter details the discretization process using the mixed finite element method and presents numerical results compared to the classical incremental procedure. The conclusion highlights the method's feasibility and suggests future research directions, including dynamic conditions and complex geometries.AI Generated
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AbstractThe ability of the symplectic Brezis-Ekeland-Nayroles principle has been shown in the standard plasticity where the material obeys associated flow rules. Recently, Harakeh et al. in Symplectic bipotentials [24] proposed a generalization of the SBEN principle to non-associated dissipative laws by replacing in the BEN functional the sum of the dissipation potential and its Fenchel polar by the bipotential. The aim of the present work is to demonstrate the capability of the extended version of the SBEN principle for the numerical simulation of non-associated elasto-plastic problems. We applied this approach to the thick-walled tube problem in the quasi-static case by using the Drucker-Prager model based on the bipotential. The accuracy of the SBEN principle is assessed by comparing the numerical results with those obtained by using the classical incremental finite element procedure. -
Collapse Load Prediction of Human Femur by Computed Tomography Based Limit Analysis
Paolo Fuschi, Cristina Falcinelli, Aurora Angela PisanoThis chapter delves into the application of Limit Analysis (LA) theory combined with Computed Tomography (CT) images to predict the collapse load of the human femur, a critical factor in assessing fracture risk. The study addresses the shortcomings of traditional diagnostic methods, which rely on areal bone mineral density (aBMD) measured by Dual-energy X-ray Absorptiometry (DXA), and proposes a more accurate approach. The chapter outlines the use of CT images for reconstructing the femur's geometry and deriving strength parameters based on bone density distribution. It introduces the Elastic Compensation Method (ECM), an iterative numerical procedure that evaluates a lower bound for the collapse load by performing sequences of linear elastic analyses. The study validates this method against experimental findings on a real femur, demonstrating its potential for clinical applications. A sensitivity analysis is conducted to assess the impact of different compressive strength-density relationships on the prediction of the collapse load. The results highlight the importance of choosing the right relationship and show that the proposed method can accurately predict the collapse load, offering a reliable tool for fracture risk assessment. The chapter concludes with a discussion on the limitations of the study and future steps to enhance the method's accuracy and applicability.AI Generated
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AbstractA limit analysis numerical procedure combined with Computed Tomography (CT) imaging is proposed to predict the collapse load of the human proximal femur. The procedure involves an accurate reconstruction of the femur geometry and a precise definition of the femur tissue strengths starting from CT images and the detected bone density using empirical relationships. A sensitivity analysis on the modeling choices is performed. The predictive capabilities of the proposed procedure are validated by comparing the numerical results with experimental findings from a real femur tested to collapse. Although the procedure is applied here to a single specimen and thus requires further refinement and investigation, the obtained results show that the method is promising and effective for providing valuable, clinically relevant information about femur fracture risk. -
Direct Estimation of the Asymptotic Cyclic Stress of Structures Under Creep and Plasticity Conditions
Konstantinos V. Spiliopoulos, Vasiliki N. TsotoulidiThis chapter delves into the complex behavior of structures subjected to extreme operational conditions, focusing on the combined effects of creep and plasticity. The primary topics covered include the introduction of an enhanced formulation of the Residual Stress Decomposition Method (RSDM) for predicting the asymptotic cyclic stress state, the theoretical foundations of the method, and its numerical implementation. The chapter also presents illustrative application examples, such as the analysis of a holed plate under mechanical load, demonstrating the effectiveness and efficiency of the procedure. The results indicate that the combined model, incorporating both creep and plasticity, exhibits a behavior that lies between the pure creep and pure plasticity models. The chapter concludes with a discussion on the method's accuracy, efficiency, and potential applications in industries where cyclic loading and creep are prevalent.AI Generated
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AbstractTo increase efficiency, mechanical engineering structures are pushed to operate under high levels of loads and temperature. Thus, they often exceed the elastic limit and develop time independent inelastic plastic strains. Also, in high temperature environment the development of time dependent inelastic creep strains is inevitable. When the loads are cyclic, with inelasticity present, the question of the assessment of the long-term strength of a structure under the continuous application of cycles can be answered by time stepping finite element procedures which follow the complete time history analysis from the very first cycle. However, a big number of cycles must be processed to reach the asymptotic long-term state. This makes the procedure, especially with creep effects present, extremely costly, as very small time steps are needed to maintain convergence. A better approach is offered by direct numerical methods that seek this asymptotic state right from the beginning of the calculations. The residual stress decomposition method (RSDM) is a successful direct method which has been developed to predict creep or plasticity asymptotic states. In the present work, the method is extended to include the combined effects of creep and plasticity. An implicit numerical scheme is presented so that convergence within a few iterations is guaranteed. The procedure can identify what kind of plastic cyclic states under creep effects is expected. At the same time, it is possible to identify the plastic-creep boundaries. A numerical example of application, under different cases of loading, underlying all the features of the procedure is presented. -
Shakedown Strength-Based Elastoplastic Topology Optimization and Its Application in Mechanical Exoskeleton Design
Songhua Huang, Zhouyi Xiang, Fuyuan Liu, Min Chen, Lele Zhang, Geng Chen, Eng Gee LimThis chapter delves into the application of shakedown strength-based elastoplastic topology optimization in the design of mechanical exoskeletons. The study begins by introducing the concept of lightweight design and its importance in reducing weight while maintaining or improving mechanical performance. It then explores the role of topology optimization in achieving this goal, focusing on the integration of shakedown theory to enhance structural performance. The methodology section provides a detailed explanation of the shakedown theory and its application in the optimization framework. It also discusses the sensitivity analysis and computational implementation of the proposed method. The numerical analysis section presents the results of applying the method to an L-shaped bracket and a mechanical exoskeleton, demonstrating the effectiveness of the approach. The study concludes with a summary of the findings and future research directions. Readers will gain insights into the advantages of incorporating shakedown analysis in topology optimization, the practical applications of the method, and the potential for further advancements in structural design.AI Generated
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AbstractTraditional structural lightweight optimization based on the elastic limit rule often leads to weight or strength redundancy, highlighting the necessity of considering elastoplastic properties for material savings. However, in typical elastoplastic topology optimization, the actual stress state of the structure must be provided, which is closely related to the loading history. In practical engineering applications, accurately describing the loading history in advance is often challenging, and only the range of load variations is typically known. Consequently, incremental elastoplastic topology optimization is impractical for real-world engineering applications. This study integrates shakedown analysis via the Direct Method with elastoplastic topology optimization. Shakedown analysis identifies a load range beyond the elastic limit but below the plastic limit, independent of loading history. The proposed method innovatively accounts for self-equilibrium residual stress at the element level, thus redefining effective and ineffective elements by replacing elastic equivalent stress with shakedown total stress. Following adjoint sensitivity analysis, the proposed method was applied to the lightweight design of a three-dimensional L-shaped bracket. This study also explores the application of this method in the design of a mechanical exoskeleton. The two cases demonstrate that our approach effectively balances the trade-off between shakedown strength and structural stiffness. These findings underscore the potential of the method and the advantage of redefining effective and ineffective elements using shakedown stress in topology optimization. -
Application of Direct Methods to Fatigue Performance: Crystal Plasticity Enhanced Direct Cyclic Algorithm and Classic Shakedown Theorem
Xuemei Lyu, Shengzhen Xin, Felix Weber, Alexander Bezold, Christoph BroeckmannThis chapter delves into the application of direct methods to enhance fatigue performance in additive manufacturing, specifically focusing on the crystal plasticity-enhanced direct cyclic algorithm (DCA) and the classic shakedown theorem (CST). The study involves the generation of statistically equivalent representative volume elements (SERVEs) based on the grain features of PBF-LB/M 316L stainless steel, incorporating defects to simulate real-world conditions. Micropillar compression tests were conducted to obtain critical resolved shear stress (CRSS) values, which were used to assign initial yield stress to each grain in the SERVEs. The chapter compares the shakedown limits predicted by DCA and CST, highlighting the differences in stress distribution and the impact of incorporating defects. The results show that CST predicts higher shakedown limits than DCA, attributed to the more homogeneous stress distribution in the grains. The study concludes that incorporating grain-specific physical properties in fatigue modeling is crucial for accurate predictions, especially in additive manufacturing. The chapter provides valuable insights into the application of direct methods for predicting fatigue performance in heterogeneous materials, offering a comprehensive analysis that is both detailed and accessible to professionals in the field.AI Generated
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AbstractAdditive manufacturing (AM) has recently been widely developed. The highly microstructural-dependent mechanical properties of the AM metallic materials are essential for their application, especially the fatigue performance during cyclic loadings. Traditional finite element method based on an incremental analysis makes it difficult to overcome expensive computations of cyclic responses for a polycrystalline alloy. However, applying direct methods to predict the stable cyclic response and the endurance limit of a polycrystalline alloy shows great computational efficiency. In this work, two direct methods, crystal plasticity enhanced direct cyclic algorithm (DCA) and von Mises yield criterion based classic shakedown theorem (CST) are applied to predict the endurance limit of laser powder bed fusion (PBF-LB/M) 316L stainless steel (SS). Fifteen statistically equivalent representative volume elements (SERVEs) were generated based on the features of the grains. The endurance limits solved by CST are 5.1–34.6% higher than those solved by DCA and the CST results show a greater scatter due to the weak stress localisation at the grain interfaces. Statistically, incorporating defects in SERVEs reduced the shakedown limits. The results indicate that incorporating grain-specific physical properties into fatigue modelling improves the reliability of local mechanical response predictions at the mesoscale. -
Limit Analysis of Metal Beams and Frames Considering Tangential Stresses and Warping
Domenico Magisano, Leonardo Leonetti, Giovanni GarceaThis chapter delves into the limit analysis of metal beams and frames, focusing on the consideration of tangential stresses and warping. It introduces a refined fiber beam-column element model that accurately captures the interaction among all stress components, including axial force, bending moments, shear forces, and torsion. The model utilizes a preliminary finite element cross-section analysis based on the generalized De Saint Venant elastic problem to establish accurate 3D elastic strains across the section. This approach improves accuracy and reliability in the inelastic range, especially for structures not dominated by flexure. The chapter also presents a mixed finite element formulation that allows for accurate results with a minimal mesh and includes non-uniform warping effects. Numerical tests validate the proposal against more detailed simulations based on solid finite elements, demonstrating the model's effectiveness in predicting collapse loads and stress distributions. The text concludes by emphasizing the importance of considering tangential stresses and warping in the inelastic response of steel members, providing a robust approach for practical applications in structural engineering.AI Generated
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AbstractThe fiber model evaluates the normal stress at a number of points over the section for a given strain increment following the plane section assumption and, by integration, axial force and bending moments. The interaction with tangential stresses is usually neglected at the point level due to the inaccurate tangential strains of the kinematics. This work proposes a generalization of the fiber model able to capture automatically the interaction among all stress components. A preliminary cross-section analysis based on the Saint Venant problem provides an accurate 3D strain as a function of the section generalized strains. This field, accurate also in the inelastic case, is exploited to impose at each section point a 3D von Mises elasto-plastic law, obtaining by integration all the resultants and moments with a full interaction. Non-uniform warping is also easily included. The section model is implemented in a mixed 3D beam-column finite element with equilibrated stress field, accurate with a minimal mesh. A suitable dual decomposition algorithm together with a proximal point strategy is used to solve the static limit analysis problem. Numerical tests show the excellent prediction of the proposal compared to analytical and solid FEM solutions also for structures not flexure-dominated. Its efficiency, on the same order as a standard fiber model, makes the approach suitable also for large buildings. -
Failure Analysis and Regularized Fracture Models
Nunziante Valoroso, Gabriele CricrìThis chapter delves into the complexities of modeling and simulating failure mechanisms and ultimate load-carrying capacity of structures, focusing on quasi-brittle fracture in concrete and other materials. It revisits Griffith's fracture criterion and explores the global energy minimization approach for studying crack evolution in two-dimensional linear elastic continua. The text introduces two fracture potentials derived from a global stationarity condition, which are computed using domain integrals. It also compares this approach with a graded damage model, highlighting their capabilities through a numerical example involving an L-shaped concrete panel. The study reveals that while the global energy approach provides coherent crack paths, the graded damage model better captures the experimental load-displacement response. The chapter concludes by discussing the complementary nature of these methods and their potential for improving the modeling of quasi-brittle materials.AI Generated
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AbstractWe present a comparison between a global energy approach à la Griffith and a gradient-regularized damage formulation named graded damage for simulating quasi-static fracture in brittle and quasi-brittle solid materials. Though intrinsically different, the two methods share a number of common features emanating from the underlying potential structure. In the Griffith-like approach the fractured state of a solid is obtained from incremental stationarity of the total free energy resulting from the sum of the bulk elastic energy and the surface energy required for crack advancement. On the other side, the graded damage model inherits the variational properties of generalized standard models, whereby the solution of the evolution problem amounts to minimize incrementally a global energy potential with convex inequality constraints. A numerical example that refers to a typical benchmark problem is presented that allows showing the features of the two approaches and their ability to reproduce experimental results as well.
- Title
- Advances in Direct Methods for Limit States of Structures and Materials
- Editors
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Konstantinos V. Spiliopoulos
Dieter Weichert
- Copyright Year
- 2026
- Publisher
- Springer Nature Switzerland
- Electronic ISBN
- 978-3-032-09203-8
- Print ISBN
- 978-3-032-09202-1
- DOI
- https://doi.org/10.1007/978-3-032-09203-8
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