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.

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.

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.

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。

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.

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.

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.

储能库井筒水泥环最大载荷解析解研究

水泥环作为地下储能库工程井筒结构的薄弱部分承受着周期载荷,对井筒的安全性有重要影响。为研究注采井筒水泥环的最大承载能力,基于极限分析中的安定理论对井筒模型进行应力分析,得到交变应力作用下的水泥环最大载荷解析解max P_(inner)。结果表明:水泥环最大载荷的解析解max P_(inner)与水泥石的内聚力、内摩擦角、井筒几何尺寸和地层地应力等因素有关,工程师可据此设计特定的参数组合,在保证井筒正常运行的同时提高压裂效率,从而满足生产要求。

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.

结构弹塑性安定分析方法

提出了一种适应予理想弹塑性结构安定分析的逐步渐近的线性规划有限元数值方法。在安定极限载荷下限因子的每步计算中，引入自平衡力系作为主子空间和暂时的局部线性化的屈服条件，大大减少了问题的变量数和约束数。基于安定分析的机动学定理，建立了一种估计安定极限载荷上限因子的方法，用于估计误差范围。通过算例进行了计算和分析，结果表明了本文方法是一种切实可行的结构安定分析数值方法。

抗剪强度随干湿循环变化对边坡安定性的影响

干湿循环将会导致岩土介质的抗剪强度产生循环变化。本文探讨了干湿循环中材料参数的循环变化对边坡安定性的影响。将材料性质的循环变化等效成载荷,并与孔隙水压的循环变化一同在安定分析中统一进行分析。最后对一算例进行计算,结果表明,即使只考虑抗剪强度参数10%的循环变化,边坡的安定安全系数也会比极限安全系数有相当程度的减小,表明抗剪强度随干湿循环的变化,会对边坡的安定性产生不可忽视的影响。

基于圆筒模型的热障涂层安定分析

热载荷作用下,由于热障涂层(thermal barrier coatings,TBCs)各层材料的热不匹配以及材料参数的温度相关等因素,会使热障涂层界面区域存在复杂的应力应变场,影响系统安定性,并导致涂层开裂和剥落.将热障涂层外凸和内凹微观界面结构简化为多层圆筒模型,借助经典机动安定定理,利用特雷斯卡(Tresca)屈服准则和增量破坏准则处理对时间的积分问题,避免了常规安定性分析的数学规划问题,建立了热障涂层安定极限分析方法,将材料屈服强度随温度变化关系简化为双线性关系,利用补偿变换的方法简化求解过程,对典型热障涂层安定性进行了研究.结果表明,利用基于圆筒的安定极限分析方法,能够方便求解安定极限,便于工程应用;热障涂层安定极限值明显高于弹性设计值,且界面外凸区域安定极限高于内凹区域极限值,结构首先在内凹处失效;圆筒模型基体曲率和涂层厚度越大,结构安定极限越高,分析结果与试验结果一致;所建立的热障涂层安定分析方法,对进一步研究考虑蠕变因素影响的热障涂层安定性具有重要意义.

基于Von-Mises屈服准则的结构安定性研究

基于静力安定定理提出了一种服从Von-mises屈服准则的静力安定分析方法。该方法将经典的安定理论与有限元方法相结合,通过构造自平衡的残余应力场,并利用有限元程序求解弹性应力场,避免了线性规划方法的维度障碍。将车辆与路面的接触应力简化为三种荷载分布形式:Hertz荷载、均布荷载和梯形荷载,研究了荷载形式和路面材料参数对结构安定行为的影响。研究结果表明,路面摩擦系数和材料屈服应力对结构安定极限的影响较大,摩擦系数越大,安定极限值越小;安定极限随着材料的屈服应力基本呈线性增长。当摩擦系数相对较小时,Hertz荷载下的安定极限值最大,而梯形分布荷载下的安定极限值最小。研究结果可为路面结构的设计提供一定的参考价值。

增量破坏型机动安定性分析方法及其应用

根据避免累积塑性变形发生的安定准则,构造了相应的安定性数值求解方法,进而用于确定海洋平台地基的安定荷载上限近似解,为评价地基在复杂变值荷载作用下的承载力提供了依据。

干湿交替对边坡长期安全性的影响

在自然条件下，边坡体内地下水位以上的部分岩土体经常处于干湿交替状态，这对其长期的安全性十分不利。在边坡所受到的全部或部分载荷发生往复变化的情况下，塑性理论要求采用安定分析来评价其安全性，而不是人们常用的、只适用于比例加载的极限分析。本文根据率形式的Koiter机动安定定理，提出了考虑地下水位往复变化情况的边坡安定上限分析方法。利用抗剪强度参数折减定义的安全系数，定量地计算载荷变化对边坡稳定性的影响。分析结果表明，干湿交替对边坡的长期安全性有较大的影响。

机动安定定理在路面结构安定分析中的应用

为了准确评价路面结构的承载能力,基于机动安定理论提出一种应用于路面结构的上限安定分析方法,计算分析荷载分布方式、路面厚度、路面材料内摩擦角及结构层间模量比对路面结构安定上限荷载影响,并与弹性极限、塑性极限及安定下限解进行对比分析.结果表明:单轮轴载下安定上限荷载随反映荷载的分布形式的a/r值增大而减小,双轮轴载下安定上限荷载随a/r值增大而增大;安定上限荷载随路面结构层厚度、材料内摩擦角增大而增大,并在面层厚度与荷载当量半径比h/r=2.5处开始趋于稳定;安定上限荷载随上下层弹性模量比E/E0的增加而减小,且在E/E0为1.0时安定上限荷载可取得最大值;路面结构安定上限荷载要大于弹性极限和安定下限荷载但小于塑性极限;安定上限荷载能较好地反映路面结构承载力,可为路面结构设计提供参考.

不同应变强化模型下结构安定性的研究

变载作用下弹塑性结构的安定性研究是一个有实际意义的课题，而结构的安定性与材料的强化规律有很大关系。本文对几种典型的强化模型下的结构安定性进行了研究，给出了在随动强化和组合强化条件下结构实现安定的条件，所得结论简化了应变强化条件下结构的安定分析。

含缺陷轴对称体的安定与极限分析

利用静力安定定理得到了计算轴对称体安定与极限载荷的统一格式，采用温度参数法构造安定分析所需的残余应力场．为了克服对工程实际问题进行安定分析时解题规模与计算精度的矛盾，针对轴对称体的特点，采用两种线性化方案对屈服面进行线性化处理，即直接内接法和在降维的应力偏量空间中对屈服面的线性化处理，使安定分析转化为一线性规划问题．在简化过程中合理选择线性化方案以便使应力校核点接近精确的屈服面；为了减小计算量，在求解过程中直接求其对偶规划．对于退化问题的数值困难采用了摄动方法来克服．本文的方法与程序通过算例验证是可靠的，并利用此方法计算了含凹坑和气孔缺陷球壳的安定与极限载荷，初步得到了含缺陷球壳的塑性承载能力与缺陷参数的关系．

Von-Mises半无限空间结构安定理论研究

进行了Hertz载荷作用下Von-Mises半无限空间的安定性研究。基于静力安定定理,通过寻找最佳残余应力场建立了平面应变条件下严格的安定极限解析方法。避免了以往安定分析方法的大型数学规划的困难,摆脱了计算中的维数障碍。研究结果表明:严格的安定极限值随着材料屈服应力的提高而增大,但随着摩擦系数的增大而减小;临界残余应力场全部位于两个界限以内,并且相交于临界点。当摩擦系数较小时,临界点发生在表面以下,随着摩擦系数逐渐增大到0.3,临界点从下层逐渐上移到表面。研究成果对安定理论应用于高速铁路设计具有参考意义。