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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Evaluation of Human Bones Load Bearing Capacity with the Limit Analysis Theory

Abstract
The present study investigates on the possibility of applying the Limit Analysis structural theory to predict a lower bound to the peak/collapse load of human bones. Such a prediction can be useful to prevent skeletal diseases, osteoporosis and bones fractures; a problem of great interest in biomechanics and of relevant socio-economic impact in modern societies. A constitutive model of Tsai-Wu-type in principal stress space is assumed for the human bone modelled in 3D and viewed, at a macroscopic level, as a structural element made of a composite anisotropic material. Simple numerical tests on in-silico idealized specimens of human femur are performed, analyzed and critically discussed.
Aurora Angela Pisano, Paolo Fuschi

### The Linear Matching Method and Its Software Tool for Creep Fatigue Damage Assessment

Abstract
The Linear Matching Method (LMM) is a numerical procedure that has undergone extensive research and development over a number of years to conduct various structural integrity assessments, more recently, the creep-fatigue damage assessment considering full creep-cyclic plasticity interaction using the extended Direct Steady Cycle Analysis. In order to encourage the widespread implementation of the LMM throughout the industry, an Abaqus CAE plug-in has been developed that enables its use by individuals with little or no understanding of the numerical theories involved. This chapter discusses different creep-cyclic plasticity mechanisms and provides a detailed review of the latest developments within the LMM framework for its evaluation. Case studies are included to demonstrate the applicability of LMM in the evaluation of creep-cyclic plasticity response for complicated loads, varying dwell periods and multi-material structures. Further, the flexibility of LMM to couple with Reversed Plasticity Domain Method to design cyclic load levels, and with design codes for creep-fatigue damage evaluation is also presented. All the results from the case studies demonstrate the level of accuracy, efficiency and robustness of the LMM.
Manu Puliyaneth, Graeme Jackson, Haofeng Chen, Yinghua Liu

### Limit Analysis of Complex 3D Steel Structures Using Second-Order Cone Programming

Abstract
The modelling of complex steel structures under static loading using rigid perfectly plastic material is presented within the framework of second-order cone programming (SOCP). The classic upper and lower bound principles of yield analysis, naturally written as optimization problems, are formulated as a pair of dual second-order cone programs which are then solved using a state-of-the art primal-dual interior point method (IPM). The IPM shows good robustness and efficiency along with reduced computational times especially for limit analysis. The whole process is illustrated first with basic steel structures checks of fillet welds or beams under biaxial bending moment, and second with complex 3D steel assemblies. The results show good agreement with the failures modes and resistance values presented in the Eurocode and allows us to obtain a reliable estimate of the ultimate resistance within a reasonable time.
Chadi El Boustani, Jeremy Bleyer, Karam Sab

### Limit Fire Analysis of 3D Framed Structures Based on Time-Dependent Yield Surfaces

Abstract
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.
Domenico Magisano, Francesco Liguori, Leonardo Leonetti, Giovanni Garcea

### Limit Analysis of Dry Masonry Block Structures with Non-associative Coulomb Friction: A Novel Computational Approach

Abstract
The limit analysis of dry-masonry block structures with non-associative Coulomb friction is formulated as a Mixed Complementarity Problem. After highlighting some of its peculiar features, such as the lack of uniqueness of the collapse multiplier, a fixed-point based algorithm is presented for constructing a solution, obtained by iteratively solving straightforward associative limit analysis problems. Supported by the comparison with benchmark problems, the resulting procedure is proven to be able to predict the collapse multiplier of masonry block structures with accuracy, robustness and effectiveness.
Nicola A. Nodargi, Claudio Intrigila, Paolo Bisegna

### Homogenization of Ductile Porous Materials by Limit and Shakedown Analysis

Abstract
This paper is a survey of recent trends in the poroplasticity combined with Direct methods. Using the hollow sphere model as Reference Elementary Volume (REV) with a matrix obeying von Mises microscopic plastic yield criterion, a stress variational model (SVM), dual of Gurson’s one, has been proposed to find by the Limit Analysis a macroscopic criterion depending on the porosity. Remarkably, it depends on the third invariant $$J_3$$ but only through its sign. Applying the normality law to the macroscopic criterion, the evolution of porosity with respect to the stress triaxiality exhibit clear discrepancies with Gurson’s one which is known to overestimate the variation of the porosity. Some extensions has been proposed to obtain a continuous dependence with respect to $$J_3$$ through Lode’s angle, to improve the strength value for the pure deviatoric loading. Thanks to the bipotential formulation, a macroscopic yield criterion was also proposed for a non associated Drucker-Prager matrix. Using the Shakedown Analysis, the method has been extended to the repeated variable loadings to obtain a fatigue criterion for the porous materials. It depends on the porosity but also strongly on Poisson’s coefficient. The general case involving shear effects with any cyclic load fluctuations ranging from the pulsating load to the alternating one is considered. The macroscopic criteria depend on the first and second macroscopic stress invariants and the sign of the third one.
Zhang Jin, Abdelbacet Oueslati, Wanqing Shen, Géry de Saxcé

### Recent Updates of the Residual Stress Decomposition Method for Shakedown Analysis

Abstract
Almost every structure or mechanical component is exposed to repeated loading conditions. As a result, materials exceed the elastic regime and plastic strains develop. The outcome of these loadings may be estimated either using a time-consuming step by step analysis or adopting modern Direct Methods which are capable to predict final cyclic states, like the elastic shakedown (safe state), the alternating plasticity, or the ratcheting (unsafe states). Towards this direction, the Residual Stress Decomposition Method (RSDM) was developed. The RSDM estimates the asymptotic cyclic state of a structure exposed to a given cyclic loading. The RSDM-S is based on the same theoretical background as RSDM and was developed in order to estimate the shakedown domain of a structure. Both methods have been tested for cyclic thermal and mechanical loads. In the present work, the RSDM-S is updated towards faster convergence by avoiding some unnecessary calculations and extended to account also for cyclic imposed displacements. Computational implementation was performed in an open source research oriented finite element analysis program. Three-dimensional brick elements are used to deal model complex geometries. The material adopted is elastic perfectly plastic von Mises type of law. Examples of application are given, proving the versatility of the approach.
Ioannis A. Kapogiannis, Konstantinos V. Spiliopoulos

### Stress Compensation Method for Shakedown Analysis and Its Engineering Applications

Abstract
This paper introduces a recently proposed direct method, the so-called stress compensation method (SCM), for shakedown analysis of engineering structures under variable repeated mechanical and thermal loads. Instead of establishing the mathematical programming formulation, the SCM performs a two-level iterative procedure based on a series of linear finite element (FE) solutions. By adding an extra stress (named the compensation stress) to the yield regions which may occur at every load vertex of the given loading domain to adjust the total stress to the yield surface and re-solving the equilibrium equations, the residual stress field for static shakedown analysis is constructed. An effective and robust iteration control scheme is presented to check the change of the compensation stress in the inner loop and to update the shakedown load multiplier in the outer loop. The numerical scheme of this method is successfully implemented into the Abaqus platform, which makes it become a general utility tool for shakedown analysis of complex structures. Numerous examples related to pressure vessel and power plant engineering are presented to illustrate the performance of the method for shakedown analysis of large-scale engineering structures under multi-dimensional loading domain.
Heng Peng, Yinghua Liu, Haofeng Chen

### On Cyclic Steady States and Elastic Shakedown in Diffusion-Induced Plasticity

Abstract
This chapter is devoted to media in which plasticity and diffusion are coupled, such as electrode materials in lithium ion batteries. We present some recent results on the large time behavior of such media when they are submitted to cyclic chemo-mechanical loadings. Under suitable technical assumptions, we notably show that there is convergence towards a cyclic steady state in which the stress, the plastic strain rate, the chemical potential and the concentration of guest atoms are all periodic in time (with the same period as the applied loading). A special case of interest is that of elastic shakedown, which corresponds to the situation where the medium behaves elastically in the large time limit. We present general theorems that allow one to construct both lower and upper bounds of the set of loadings for which elastic shakedown occurs, in the spirit of Melan and Koiter theorems in classical plasticity. An illustrative example—for which all the relevant calculations can be done in closed-form—is presented.
Michaël Peigney

### Numerical Method for Quasi-static and Dynamic Elastoplastic Problems by Symplectic Brezis-Ekeland-Nayroles Non-incremental Principle

Abstract
Most computer-aided engineering software provide a classical incremental computation procedure for nonlinear problems. Although little used in the literature, the Brezis-Ekeland-Nayroles (BEN) principle, an alternative step-by-step algorithm, based on the time integration of the sum of the dissipation potential and its Fenchel polar can have a global view of whole evolution. In short, the BEN principle converts a mechanical problem to a constrained optimization problem. Recently, Buliga and de Saxcé have proposed a symplectic version of the BEN principle which generalizes the Hamiltonian inclusion formalism for the dissipative systems. In the present work, this formalism is specialized to the standard plasticity in small, finite strains, in statics and dynamics. We apply it numerically to solve the classical problem of a tube problem in plane strain subjected to an internal pressure in statics and dynamics. An excellent agreement is obtained between the numerical results obtained by the BEN approach and the reference numerical solution.
Xiaodan Cao, Abdelbacet Oueslati, An Danh Nguyen, Marcus Stoffel, Bernd Market, Géry de Saxcé

### Shakedown Limits of Slab Track Substructures and Their Implications for Design

Abstract
This paper presents an approach to shakedown of slab track substructures subjected to train loads. The train load is converted into a distributed moving load on the substructure surface using a simplified track analysis. Based on the lower-bound dynamic shakedown theorem, shakedown solutions for the slab track substructures are obtained over a range of train speeds between zero and the critical speed of the track. It is found the shakedown limit is largely influenced by the ratio of layer elastic moduli and the ratio of train speed to critical speed rather than their absolute values. An attenuation factor, as a function of the critical speed and the friction angle of subsoil, is proposed to effectively obtain the shakedown limit of the slab track substructure at any train speed. In light of the shakedown solutions, improvements to the existing design and analysis approaches are also suggested.
Juan Wang, Hai-Sui Yu, Shu Liu

### Investigations of Shakedown in the Presence of Ambient Creep Using Direct Methods for High Strength Steel Under Multiaxial Loadings

Abstract
Life integrity assessment of industrial components often requires investigations of the cyclic inelastic response at a range of operating temperatures. Some high strength steels exhibit a well-known ambient temperature creep behaviour, which can also impact the cyclic behaviour, especially under long-term operation. In this study, a direct method known as the Linear Matching Method has been used to predict the cyclic shakedown and ratchet limits of high-strength steel (AISI 1144). The numerical predictions are compared with a recent testing campaign that was completed at room temperature to characterise the multiaxial behaviour of AISI 1144. Due to creep of the material, inelastic strain accumulation is also observed for loading conditions within the shakedown limit. The extended Direct Steady Cyclic Analysis (eDSCA) approach has been used to predict the cyclic behaviour in the presence of creep. In addition, for specific load cases of interest, a newly revised creep-ratcheting limit has been derived and compared with the experimental tests.
Daniele Barbera, Ali Charbal, I. Soner Cinoglu, Natasha Vermaak
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