Effect of boundary conditions on shakedown analysis of heterogeneous materials

The determination of the ultimate load-bearing capacity of structures made of elastoplastic heterogeneous materials under varying loads is of great importance for engineering analysis and design. Therefore, it is necessary to accurately predict the shakedown domains of these materials. The static shakedown theorem, also known as Melan's theorem, is a fundamental method used to predict the shakedown domains of structures and materials. Within this method, a key aspect lies in the construction and application of an appropriate self-equilibrium stress field(SSF). In the structural shakedown analysis, the SSF is typically constructed by governing equations that satisfy no external force(NEF) boundary conditions. However, we discover that directly applying these governing equations is not suitable for the shakedown analysis of heterogeneous materials. Researchers must consider the requirements imposed by the Hill-Mandel condition for boundary conditions and the physical significance of representative volume elements(RVEs). This paper addresses this issue and demonstrates that the sizes of SSFs vary under different boundary conditions, such as uniform displacement boundary conditions(DBCs), uniform traction boundary conditions(TBCs), and periodic boundary conditions(PBCs). As a result, significant discrepancies arise in the predicted shakedown domain sizes of heterogeneous materials. Built on the demonstrated relationship between SSFs under different boundary conditions, this study explores the conservative relationships among different shakedown domains, and provides proof of the relationship between the elastic limit(EL) factors and the shakedown loading factors under the loading domain of two load vertices. By utilizing numerical examples, we highlight the conservatism present in certain results reported in the existing literature. Among the investigated boundary conditions, the obtained shakedown domain is the most conservative under TBCs.Conversely, utilizing PBCs to construct an SSF for the shakedown analysis leads to less conservative lower bounds, indicating that PBCs should be employed as the preferred boundary conditions for the shakedown analysis of heterogeneous materials.

南方湿热地区路基红黏土Shakedown临界应力水平试验研究

根据南方湿热地区气候特点和10条典型高速公路路基受力计算,制定路基红黏土的重复荷载动三轴试验方案;研究不同压实度、含水率、轴向应力水平、循环加载次数等条件下路基红黏土的塑性力学行为,并结合Shakedown概念,界定循环动荷载作用下红黏土的Shakedown临界应力水平。研究结果表明:压实度对红黏土临界应力水平影响较小,含水率对其影响显著,最佳含水率Com及110%Com,120%Com和130%Com对应的临界应力水平分别为70%,70%,60%和40%,据此可绘制南方湿热地区路基红黏土Shakedown临界应力包络线;设计时,应以施工含水率对应的临界应力水平作为路基受力的控制上限,从而保证路基的变形稳定和耐久性。

General Analytical Shakedown Solution for Structures with Kinematic Hardening Materials

The effect of kinematic hardening behavior on the shakedown behaviors of structure has been investigated by performing shakedown analysis for some specific problems. The results obtained only show that the shakedown limit loads of structures with kinematic hardening model are larger than or equal to those with perfectly plastic model of the same initial yield stress. To further investigate the rules governing the different shakedown behaviors of kinematic hardening structures, the extended shakedown theorem for limited kinematic hardening is applied, the shakedown condition is then proposed, and a general analytical solution for the structural shakedown limit load is thus derived. The analytical shakedown limit loads for fully reversed cyclic loading and non-fully reversed cyclic loading are then given based on the general solution. The resulting analytical solution is applied to some specific problems: a hollow specimen subjected to tension and torsion, a flanged pipe subjected to pressure and axial force and a square plate with small central hole subjected to biaxial tension. The results obtained are compared with those in literatures, they are consistent with each other. Based on the resulting general analytical solution, rules governing the general effects of kinematic hardening behavior on the shakedown behavior of structure are clearly.

New shakedown criterion and permanent deformation properties of unbound granular materials

Unbound granular material specifications for road pavements in Australia are primarily based on physical material specification rather than mechanical characterisation. This simplified approach does not reflect the actual material performance under repeated dynamic traffic loads. There is a little information available on the influence of the local crushed rock properties and compacted layer properties on permanent deformation (PD). This study aims to characterise the local unbound granular materials in Victoria according to their PD behaviour under repeated loads and to develop a suitable shakedown criterion that could describe the PD of the tested materials to simplify the flexible pavement design. Repeated-load triaxial tests were conducted over several samples with a range of moisture contents, gradations, densities, and stress conditions. The laboratory test results showed that PD behaviour was influenced by several factors. In addition, the tested subbase-specified unbound granular materials reflect high PD resistance that is almost equivalent to basequality unbound granular materials. This may indicate that current requirements for the subbase-quality unbound granular materials are over-prescribe. Moreover, as the existing shakedown criterion was not applicable for the multi-stage repeated-load triaxial test and the local tested materials, a new shakedown criterion and new boundaries are proposed based on the PD behaviour. In the proposed criterion, the shakedown ranges are identified based on the curve angle of the PD vs. logarithm of the number of loading cycles, and this new criterion was validated using several materials from existing literature. The local tested base and subbase materials can be assigned as Range A when PD\1%, Range B when 1%\PD\3%, and Range C when PD[3%. The proposed criterion could provide a useful and quick approach to assess the PD of the unbound granular materials with both single and multistages of stresses.

A Shakedown Strength Based Parametric Optimization Technique and Its Application on an Airtight Module

In aerospace engineering,design and optimization of mechanical structures are usually performed with respect to elastic limit.Besides causing insufcient use of the material,such design concept fails to meet the ever growing needs of the light weight design.To remedy this problem,in the present study,a shakedown theory based numerical approach for performing parametric optimization is presented.Within this approach,strength of the structure is measured by its shakedown limit calculated from the direct method.The numerical method developed for the structural optimization consists of nested loops:the inner loop adopts the interior point method to solve shakedown problems pertained to fxed design parameters,while the outer loop employs the genetic algorithm to fnd optimal design parameters leading to the greatest shakedown limit.The method established is frst verifed by the classic plate-with-a-circular-hole example,and after that it is applied to an airtight module for determining few key design parameters.By carefully analyzing results generated during the optimization process,it is convinced that the approach can become a viable means for designing similar aerospace structures.

General Analytical Shakedown Analysis of a Structure Subjected to Combined Loading with Constant and Cyclic Loads

The shakedown behavior of structures subjected to a combined loading of constant and cyclic loads has been well researched.For some specified problems,shakedown limit loads have been obtained.However,the general effect of combined loading on the structural shakedown has not yet been presented.The general analytic solution of the elastic shakedown limit load is thus derived for a structure subjected to combined loading.Polizzotto's extended static shakedown theorem for combined loading is applied.The stress field in equilibrium with the external constant load required in Polizzotto's extended theorem is constructed by subtracting the reference elastic stress field of the peak cyclic load from the elastic-plastic stress field of the combined constant load and peak cyclic load.The shakedown condition of the stress field is then derived according to the extended theorem.Through the analytical analysis of the shakedown condition,the structural shakedown behavior under combined loading is investigated.A general solution of the shakedown limit load is then derived,and the effects of the combined loading on the shakedown behavior are proposed.The obtained general analytical result is applied to a hollow tension specimen under constant tension and alternating torsion and a plate with a central hole under constant and cyclic tension.The results are consistent with the solutions reported in the literature.

基于shakedown原理的路面设计方法

主要介绍了一种可以用来分析无结合料路面各种破坏机理的数学模型,与认为路面从一开始就受循环荷载作用的现有模型不同,该模型采用shakedown原理来替代基于基础分析的现有方法。引入了该设计方法的基本概念,研究了车辙和表层下面滑动引起的破坏,并论述了如何将其应用于设计过程中。

Shakedown Criterion Employing Actual Residual Stress Field and Its Application in Numerical Shakedown Analysis

Construction of the static admissible residual stress field and searching the optimal field are key tasks in the shakedown analysis methods applying the static theorem. These methods always meet dimension obstacles when dealing with complex problems. In this paper, a novel shakedown criterion is proposed employing actual residual stress field based on the static shakedown theorem. The actual residual stress field used here is produced under a specified load path, which is a sequence of proportional loading and unloading from zero to all the vertices of the given load domain. This ensures that the shakedown behavior in the whole load domain can be determined based on the theorem proposed by K?nig. The shakedown criterion is then implemented in numerical shakedown analysis. The actual residual stress fields are calculated by incremental finite element elastic-plastic analysis technique for finite deformation under the specified load path with different load levels. The shakedown behavior and the shakedown limit load are determined according to the proposed criterion. The validation of the criterion is performed by a benchmark shakedown example, which is a square plate with a central hole under biaxial loading. The results are consistent with existing results in the literatures and are validated by full cyclic elastic-plastic finite element analysis. The numerical shakedown analysis applying the proposed criterion avoids processing dimension obstacles and performing full cyclic elastic-plastic analysis under arbitrary load paths which should be accounted for appearing. The effect of material model and geometric changes on shakedown behavior can be considered conveniently.

SIMPLE SHAKEDOWN OF STRUCTURES UNDER VARIABLE MULTI-LOADINGS

在可变 multi-loadings 下面的结构的临时的便床分析被考虑，并且相应简单临时的便床条件在这份报纸被介绍。固定压力地的分发被给，并且固定压力地的自我平衡被分析。有弹性的临时的便床和塑料临时的便床条件基于固定压力地被介绍。定理是方便的多在轮转的变量下面评估结构的临时的便床限制 -- 通过在结构的一系列危险位置解决积极的分级的域和固定压力域因素的装载，和乏味的计算被避免。最后，定理在变量下面被用于一个厚墙的圆柱的试管压力和温度，和滚动接触问题。结果在对一些计算结果的好同意。

A SOLUTION PROCEDURE FOR LOWER BOUND LIMIT AND SHAKEDOWN ANALYSIS BY SGBEM

The symmetric Galerkin boundary element method (SGBEM) instead of the finite element method is used to perform lower bound limit and shakedown analysis of structures. The self-equilibrium stress fields are constructed by a linear combination of several basic self-equilibrium stress fields with parameters to be determined. These basic self-equilibrium stress fields are expressed as elastic responses of the body to imposed permanent strains and obtained through elastic-plastic incremental analysis. The complex method is used to solve nonlinear programming and determine the maximal load amplifier. The limit analysis is treated as a special case of shakedown analysis in which only the proportional loading is considered. The numerical results show that SGBEM is effcient and accurate for solving limit and shakedown analysis problems.

Shear Effects in Shakedown Analysis of Offshore Structures

In this paper, a formulation for shakedown analysis of elastic-plastic offshore structures under cyclic wave loading is presented. In this formulation, a fast numerical solution method is used, suitable for the Finite Element Method (FEM) analysis of large offshore structures on which shear effects in addition to bending and axial effects are taken into account. The Morison equation is adopted for converting the velocity and acceleration terms into resultant forces and it is extended to consider arbitrary orientations of the structural members. The theoretical methods of the shakedown analysis are discussed in detail and the formulation is applied to an offshore structure to verify the concept employed and its analytical capabilities.

ANALYSIS OF SHAKEDOWN OF FG BREE PLATE SUBJECTED TO COUPLED THERMAL-MECHANICAL LOADINGS

The static and kinematic shakedown of a functionally graded (FG) Bree plate is analyzed. The plate is subjected to coupled constant mechanical load and cyclically varying temperature. The material is assumed linearly elastic and nonlinear isotropic hardening with elastic modulus,yield strength and the thermal expansion coeffcient varying exponentially through the thickness of the plate. The boundaries between the shakedown area and the areas of elasticity,incremental collapse and reversed plasticity are determined,respectively. The shakedown of the counterpart made of homogeneous material with average material properties is also analyzed. The comparison between the results obtained in the two cases exhibits distinct qualitative and quantitative difference,indicating the importance of shakedown analysis for FG structures. Since FG structures are usually used in the cases where severe coupled cyclic thermal and mechanical loadings are applied,the approach developed and the results obtained are significant for the analysis and design of such kind of structures.

曲线线路重载机车轮缘裂纹损伤及影响因素研究

针对我国某型大功率重载机车在高原多曲线线路上运行时频繁出现轮缘裂纹的问题,采用理论分析、数值模拟与现场试验相结合的方法对机车轮缘裂纹损伤进行研究。首先,基于多体动力学和安定理论,建立了该型大功率重载机车车辆动力学模型及轮缘裂纹损伤预测模型:其次,通过现场测试数据与数值仿真结果对比,验证了模型的正确性;最后,根据该型重载机车运行线路特点,分析了曲线半径、曲线超高、曲线坡度等线路参数及轮缘润滑对重载机车轮缘裂纹损伤的影响规律。研究结果表明,连续小半径曲线是该型重载机车轮缘裂纹产生的主要原因,此外轮缘润滑对机车轮缘裂纹损伤有显著影响,可通过合理的线路参数设置及轮缘润滑来降低机车轮轨动态相互作用,进而减少重载机车轮缘裂纹的发生。

基于SIMP-安定的高速列车组合式座椅底架轻量化设计与分析

在高速列车座椅轻量化设计中,传统设计方法所普遍采用的结构优化技术仅以结构刚度或局部应力水平为目标,无法得到交变载荷作用下效率最高的结构构型.列车座椅在实际使用中受到轨道与应用场景的影响,除主结构需具备一定强度外,其关键部件也要求在交变载荷作用下不发生明显变形.在此背景下,提出了一种采用固体各向同性惩罚微结构插值(SIMP)算法对主结构进行以刚度为目标的拓扑优化,同时以安定直接法对关键部件进行以安定强度为目标的参数优化一体化的设计与分析方法.采用所提出的方法对高速列车组合式座椅底架进行了优化,取得了显著成果.优化后的座椅底架在性能满足要求的前提下,底架主结构减重17%、L型连接件承载交变载荷的结构效率提高23%.所提出的研究方法及结果对于同类结构的轻量化设计具有重要意义.

Upper Bound Shakedown Analysis of Plates Utilizing the C^(1) Natural Element Method

This paper proposes a numerical solution method for upper bound shakedown analysis of perfectly elasto-plastic thin plates by employing the C^(1) natural element method.Based on the Koiter’s theorem and von Mises yield criterion,the nonlinear mathematical programming formulation for upper bound shakedown analysis of thin plates is established.In this formulation,the trail function of residual displacement increment is approximated by using the C^(1) shape functions,the plastic incompressibility condition is satisfied by introducing a constant matrix in the objective function,and the time integration is resolved by using the Konig’s technique.Meanwhile,the objective function is linearized by distinguishing the non-plastic integral points from the plastic integral points and revising the objective function and associated equality constraints at each iteration.Finally,the upper bound shakedown load multipliers of thin plates are obtained by direct iterative and monotone convergence processes.Several benchmark examples verify the good precision and fast convergence of this proposed method.

不同建筑固废再生骨料取代率下粗粒土填料永久变形特性及安定行为研究

为探究城市建筑拆除固废再生骨料部分或全部取代天然骨料用于粗粒土路基填料的可行性,开展不同再生骨料取代率、含水率、围压和剪应力比等组合下的室内大型静动三轴试验,定量研究土性参数和应力状态对试样抗剪强度和累积塑性应变特性的影响规律。基于半对数坐标下累积塑性应变发展的多阶段特征,分别考虑不同阶段塑性变形累积速率以及相邻两阶段的塑性变形累积速率的差异,提出适用于建筑固废再生骨料路基填料的新型安定行为判定准则。研究结果表明:再生骨料路基填料试样的累积塑性应变随含水率和剪应力比的增大而增大,当再生骨料路基填料试样在剪应力比为0.3和0.5时,抗累积变形性能与天然骨料路基填料试验所得的抗累积变形性能接近,综合考虑抗剪强度和累积塑性变形特性的再生骨料路基填料最优取代率为85%;新安定行为判定准则具有较高的准确性,可为相似路基填料的长期路用性能评定提供理论依据。

基于耗散能的高速铁路路基填料动力安定临界阈值研究

科学评估高速铁路路基服役安定特性是保障复杂环境下路基百年服役期的耐久稳定的重要理论手段。对典型高速铁路路基粗粒土填料开展长期循环加载动三轴试验,揭示了不同围压、动载幅值等因素对填料变形发展规律的影响,基于A塑性安定、B塑形蠕变、C增量破坏3类安定状态分析了长期变形收敛趋势以及耗散能变化规律。构建了单位体积耗散能与安定状态的关联,提出了基于等效单位体积耗散能的安定临界状态判别标准和动应力、动应变阈值确定方法。本文研究的典型填料,引起高铁路基填料由稳定的安定状态向不稳定的增量破坏状态的临界动应变所处范围为0.8×10^(-3)~1.3×10^(-3)。研究结果可对涉及列车动载长期作用下的高速铁路路基安定性分析和长期稳定性评估提供理论参考。

Shakedown and Limit Analysis of Defective Pressure Vessels

This paper presents a failure analysis of pressure vessels with defects by a direct method of limit and shakedown analysis. The defects considered are part through slots with various geometric configurations. The engineering situation considered here has practical importance in the pressure vessel industry. The results are compared with those obtained by a step by step procedure using the professional code ABAQUS and where possible, with those provided by semiempirical formulae used by industry. The limit and shakedown analysis methods are found to be more economical and more reliable than marching solutions achieved by step by step elastic plastic analysis. The effects of various part through slots on the load carrying capacities of pressure vessels are investigated.

AN UPPER DISPLACEMENT BOUND IN DYNAMIC SHAKEDOWN PROBLEMS FOR ELASTOKINEMATIC WORK-HARDENING MATERIALS

A dynamic shakedown theorem for an elasto-kinematic work-hardenlng material with an arbitrary convex yield surface is reviewed. An upper displacement bound for this dynamic shakedown structure is derived. Except for the condition necessary for shakedown of the structure, no other restriction is imposed, which eliminates some of the difficulties appearing in Previous works.

Shakedown Analysis of 3-D Structures Using the Boundary Element Method Based on the Static Theorem

The static shakedown theorem was reformulated for the boundary element method (BEM) rather than the finite element method with Melan’s theorem, then used to develop a numerical solution procedure for shakedown analysis. The self-equilibrium stress field was constructed by a linear combination of several basis self-equilibrium stress fields with undetermined parameters. These basis self-equilibrium stress fields were expressed as elastic responses of the body to imposed permanent strains obtained using a 3-D BEM elastic-plastic incremental analysis. The lower bound for the shakedown load was obtained from a series of nonlinear mathematical programming problems solved using the Complex method. Numerical examples verified the precision of the present method.