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.

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 K6nig. 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 he considered conveniently.

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.

SIMPLE SHAKEDOWN OF STRUCTURES UNDER VARIABLE MULTI-LOADINGS

The shakedown analysis of structures under variable multi-loadings is considered, and the corresponding simple shakedown condition is presented in this paper. Distribution of fixed stresses field is given, and the self-equilibrium of fixed stresses field is analyzed. Elastic shakedown and plastic shakedown conditions are presented based on the fixed stresses field. The theorem is convenient to evaluate the shakedown limit of structures under cyclical variable multiloadings through solving positive scalar fields and fixed stresses field factors at a series of dangerous positions of the structure, and tedious computations are avoided. Finally the theorem is applied to a thick-walled cylindrical tube under variable pressure and temperature, and the rolling contact problem. The results are in good agreement with some computational results.

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

The symmetric Galerkin boundary element method (SGBEM) instead ofthe finite element method is used t perform lower bound limit andshakedown analysis of structures. The self-equilibrium stress fieldsare constructed by a linear combination of several basicself-equilibrium stress fields with parameters to be determined.These basic self-equilibrium stress fields are expressed as elasticresponses of the body to im- posed permanent strains and obtainedthrough elastic-plastic incremental analysis.

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.

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.

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.

SHAKEDOWN ANALYSIS OF SHELL STRUCTURES OF KINEMATIC HARDENING MATERIALS

It is of great practical importance to analyze the shakedown of shell structures under cyclic loading, especially of those made of strain hardening materials.In this paper, same further understanding of the shakedown theorem for kinematic hardening materials has been made, and it is applied to analyze the shakedown of shell structures Though the residual stress of a real stale is related to plastic strain, the time-independent residual stress field as we will show in the theorem may be unrelated to the time-independent kinematically admissible plastic strain field For the engineering application, it will lie much more convenient to point this out clearly and definitely, otherwise it will be very difficult. Also, we have proposed a new method of proving this theorem.The above theorem is applied to the shakedown analysis of a cylindrical shell with hemispherical ends. According to the elastic solution, various possible residual sfcss and plastic strain Jlelds, the shakedown analysis of the structure can be reduced to a mathematical programming problem.The results of calculation show that the shakedown load oj strain hardening materials is about 30-40% higher than that of ideal plastic materials. So it is very important to consider the hardening of materials in the shakedown analysis,for it can greatly increase the structure design capacity, and meanwhile provide ascicntific basis to improve the design of shell structures.

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。

Kinematic Shakedown Analysis for Strain-Hardening Plates with the C1 Nodal Natural Element Method

This paper proposes a novel numerical solution approach for the kinematic shakedown analysis of strain-hardening thin plates using the C^(1)nodal natural element method(C^(1)nodal NEM).Based on Koiter’s theorem and the von Mises and two-surface yield criteria,a nonlinear mathematical programming formulation is constructed for the kinematic shakedown analysis of strain-hardening thin plates,and the C^(1)nodal NEM is adopted for discretization.Additionally,König’s theory is used to deal with time integration by treating the generalized plastic strain increment at each load vertex.A direct iterative method is developed to linearize and solve this formulation by modifying the relevant objective function and equality constraints at each iteration.Kinematic shakedown load factors are directly calculated in a monotonically converging manner.Numerical examples validate the accuracy and convergence of the developed method and illustrate the influences of limited and unlimited strain-hardening models on the kinematic shakedown load factors of thin square and circular plates.

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.

长期循环荷载下黄泛区黄河泥沙的变形特性及安定性分析

随着国家“黄河高质量发展”战略的提出,沿黄工程建设飞速发展,填筑土体材料需求量剧增,建筑资源短缺的问题突显。黄河泥沙作为一种新型工程填料,其资源利用前景被相关部门和研究人员重视。为研究长期循环荷载作用下的黄河泥沙的动力服役特性,选取黄河泥沙典型试样,基于GDS三轴试验系统进行了排水条件下的大周次(10000次)循环三轴试验,结合安定理论系统分析围压、循环应力比、加载频率和密实度对黄河泥沙变形特性的影响,并对黄河泥沙的长期动力稳定性进行研究。研究结果表明:累积应变和回弹应变随着围压的增大而增大,随着频率的升高和密实度的增大而减小,循环应力比较小时,应变累积在加载初期几百次循环内就迅速稳定;随着循环应力比增大,应变的发展逐渐具有蠕变的性质,在5000次循环后仍不断累积;随着循环应力比继续增大,轴向应变将在加载初期迅速增大,泥沙试样在很少的周次内达到破坏状态。通过与代表性的相似土体进行对比,发现黄河泥沙的蠕变性介于粉土和砂砾土之间。基于黄河泥沙的安定性分析,提出了黄河泥沙容许循环应力比的表达式,用以初步判断排水状态下黄河泥沙的长期动力稳定状态。研究结果为泥沙土在实际工程中的进一步应用和推广提供了参数支撑。

压力容器及容器-管道系统的安定载荷计算方法对比

针对压力容器及容器-管道系统的安定载荷计算,介绍了循环弹塑性分析法、下限安定定理法和双极限载荷法3种求解安定载荷方法的原理和流程。基于这3种方法,具体计算了机械载荷下厚壁圆筒和标准椭圆封头中心接管的安定载荷,以及热载荷下标准椭圆封头中心管系整体模型的安定载荷,并对比分析了3种方法的优缺点。结果表明,循环弹塑性分析法能较准确地反映结构的实际承载能力,但求解复杂、工作量最大;双极限载荷法和下限安定定理法相对于循环弹塑性分析法的最大误差均小于5%,且偏保守,双极限载荷法的计算量最小。综合考虑计算精度和工作量,在工程设计中可采用双极限载荷法;对于计算精度要求较高、周围弹性材料对最大应力点的约束能力较弱的情况,建议采用循环弹塑性分析法。

寒区铁路冻结粗粒土填料长期变形特性研究

振动列车荷载长期循环往复作用下土体产生的变形存在一定程度的应变累积,依托某寒区铁路,开展不同温度下冻结粗粒土填料长期动三轴试验,探究填料长期变形、弹性模量和阻尼比的变化规律,并对试样累积塑性变形进行安定性分析,将累积塑性变形划分为三种不同的状态。在黏弹性模型的基础上串联塑性弹簧提出一种计算应力-应变关系的简化方法。研究结果表明:温度越低,动弹性模量越高,阻尼比越低。在动应力水平较低时,土体处在塑性安定阶段,累积塑性变形随振动次数的增加趋于稳定,动弹性模量略有增加,阻尼比则逐渐减小;当动应力水平超过塑性蠕变界限时,土体刚度迅速降低,累积塑性变形急剧增加,试样呈现破坏的趋势;当动应力水平处于塑性安定和塑性蠕变界限之间时,累积塑性变形虽然增加,但仍然处在可控范围。提出的弹塑性模型可以很好地描述冻结粗粒土填料的滞回特性和累积塑性变形规律。研究成果可为高速铁路路基填料长期动力性能的分析提供有益参考。

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.