Extended bounding theorems of limit analysis

This paper studies the bounding problems of the complete solu-tion of limit analysis for a rigid-perfectly plastic medium,allowing for the discontinuity of plastic flow.A generalizedvariational principle involving conditions of the rigid-plas-tic interface and the discontinuous surface of a velocityfield has been advanced for the mixed-boundary value problem.Based on this principle,a set of variational formulae of li-mit analysis is established.The safety factors obtained bythese formulae lie between the upper and lower bounds obtainedby the classical bounding theorems with the same kinematicallyand statically admissible field.Moreover,extended bounding theorems have been derivedand proved,which hold a broader stress and velocity field thanthe statically and kinematically admissible field.The corol-laries of these theorems indicate the relationship between thevariational solution and the complete solution of limit analy-sis.Applications of these theorems show that a close approxi-mation can be obtained     

Strain gradient plasticity,strengthening effects and plastic limit analysis

Within the framework of isotropic strain gradient plasticity, a rate-independent constitutive model exhibiting size dependent hardening is formulated and discussed with particular concern to its strengthening behavior. The latter is modelled as a (fictitious) isotropic hardening featured by a potential which is a positively degree-one homogeneous function of the effective plastic strain and its gradient. This potential leads to a strengthening law in which the strengthening stress, i.e. the increase of the plastically undeformed material initial yield stress, is related to the effective plastic strain through a second order PDE and related higher order boundary conditions. The plasticity flow laws, with the role there played by the strengthening stress, are addressed and shown to admit a maximum dissipation principle. For an idealized elastic perfectly plastic material with strengthening effects, the plastic collapse load problem of a micro/nano scale structure is addressed and its basic features under the light of classical plastic limit analysis are pointed out. It is found that the conceptual framework of classical limit analysis, including the notion of rigid-plastic behavior, remains valid. The lower bound and upper bound theorems of classical limit analysis are extended to strengthening materials. A static-type maximum principle and a kinematic-type minimum principle, consequences of the lower and upper bound theorems, respectively, are each independently shown to solve the collapse load problem. These principles coincide with their respective classical counterparts in the case of simple material. Comparisons with existing theories are provided. An application of this nonclassical plastic limit analysis to a simple shear model is also presented, in which the plastic collapse load is shown to increase with the decreasing sample size (Hall–Petch size effects).     

Theorems on the limit analysis of finite deformation

A generalized variational principle (theorem 1) which is equivalent mathematically to the whole set of equations and conditions and must be satisfied by the limit analysis of finite deformation is proposed in this paper. It is also proved that the limit load deduced from theorem 1 will lie between the lower and upper bounds given by the bound theorems of finite deformation     

Application of the kinematical shakedown theorem to rolling and sliding point contacts

In the plane-strain conditions of a long cylinder in rolling line contact with an elastic-perfectly-plastic half-space an exact shakedown limit has been established previously by use of both the statical (lower bound) and kinematical (upper bound) shakedown theorems. At loads above this limit incremental strain growth or “ratchetting” takes place by a mechanism in which surface layers are plastically sheared relative to the subsurface material.In this paper the kinematical shakedown theorem is used to investigate this mode of deformation for rolling and sliding point contacts, in which a Hertz pressure and frictional traction act on an elliptical area which repeatedly traverses the surface of a half-space. Although a similar mechanism of incremental collapse is possible, the behaviour is found to be different from that in two-dimensional line contact in three significant ways: (i) To develop a mechanism for incremental growth the plastic shear zone must spread to the surface at the sides of the contact so that a complete segment of material immediately beneath the loaded area is free to displace relative to the remainder of the half-space, (ii) Residual shear stresses orthogonal to the surface are developed in the subsurface layers, (iii) A range of loads is found in which a closed cycle of alternating plasticity takes place without incremental growth, a condition often referred to as “plastic shakedown”.Optimal upper bounds to both the elastic and plastic shakedown limits have been found for varying coefficients of traction and shapes of the loaded ellipse. The analysis also gives estimates of the residual orthogonal shear stresses which are induced.     

Limit analysis of anisotropic soils using finite elements and linear programming

The main purpose of this paper is to present a finite element formulation of the bound theorems which allows for the variation of soil strength with direction. To achieve this objective, the conventional isotropic Mohr-Coulomb yield criterion is generalised to include the effect of strength anisotropy. The finite element limit analysis formulation using the modified anisotropic yield criterion is then developed. Several examples are given in the paper to illustrate the capability and effectiveness of the proposed numerical procedure for computing rigorous bounds for anisotropic soils.     

Macroscopic strain potentials in nonlinear porous materials

By taking a hollow sphere as a representative volume element (RVE), the macroscopic strain potentials of porous materials with power-law incompressible matrix are studied in this paper. According to the principles of the minimum potential energy in nonlinear elasticity and the variational procedure, static admissible stress fields and kinematic admissible displacement fields are constructed, and hence the upper and the lower bounds of the macroscopic strain potential are obtained. The bounds given in the present paper differ so slightly that they both provide perfect approximations of the exact strain potential of the studied porous materials. It is also found that the upper bound proposed by previous authors is much higher than the present one, and the lower bounds given by Cocks is much lower. Moreover, the present calculation is also compared with the variational lower bound of Ponte Castañeda for statistically isotropic porous materials. Finally, the validity of the hollow spherical RVE for the studied nonlinear porous material is discussed by the difference between the present numerical results and the Cocks bound.     

Kinematic limit analysis of frictional materials using nonlinear programming

In this paper, a nonlinear numerical technique is developed to calculate the plastic limit loads and failure modes of frictional materials by means of mathematical programming, limit analysis and the conventional displacement-based finite element method. The analysis is based on a general yield function which can take the form of the Mohr–Coulomb or Drucker–Prager criterion. By using an associated flow rule, a general nonlinear yield criterion can be directly introduced into the kinematic theorem of limit analysis without linearization. The plastic dissipation power can then be expressed in terms of kinematically admissible velocity fields and a nonlinear optimization formulation is obtained. The nonlinear formulation only has one constraint and requires considerably less computational effort than a linear programming formulation. The calculation is based entirely on kinematically admissible velocities without calculation of the stress field. The finite element formulation of kinematic limit analysis is developed and solved as a nonlinear mathematical programming problem subject to a single equality constraint. The objective function corresponds to the plastic dissipation power which is then minimized to give an upper bound to the true limit load. An effective, direct iterative algorithm for kinematic limit analysis is proposed in this paper to solve the resulting nonlinear mathematical programming problem. The effectiveness and efficiency of the proposed method have been illustrated through a number of numerical examples.     

Extensions of the Prager-Shield theory of optimal plastic design

The Prager-Shield associated displacement field method for optimal plastic design is extended to multi-component specific cost functions and multiple load conditions, and a lower bound theorem based on kinematic requirements only is introduced. Since any statically admissible stress field results in an upper bound, the proposed theorem provides a simple method for establishing bounds on the optimal cost. By a simple substitution of parameters into the general equations presented, the optimality criteria can be obtained for particular design problems. Examples of optimal fibre-reinforced plates are given.     

Bornes quasi-inferieures et bornes superieures de la pression de ruine des coques de revolution par la methode des elements finis et par la programmation non-lineaire

This work is concerned with rigid-plastic limit analysis of shells of revolution subject to rotationally symmetric loadings. These shells, of arbitrary shape, are discretized into a series of finite elements, each being a conical shell. The von Mises condition, valid for sandwich shells, is used to compare the results obtained with existing values. After assembling the finite elements, the limit analysis program is reduced to a simple application of the non-linear programming technique where both the sequential unconstrained minimization technique (SUMT) and feasible conjugate direction (FCD) are utilized respectively for statically admissible and kinematically admissible approaches. Therefore, upper bounds and lower bounds of the collapse loads are found for some sample problems : conical, spherical, ellipsoidal, torispherical shells etc. These numerical results are illustrated and compared with existing ones described in the literature.     

Creep life predictions for structures subject to variable loading

Previous work which established upper and lower bounds on the creep life of steadily loaded structures is extended to cater for load and temperature variations in non-homogeneous structures. The investigation is limited to the range where short term plasticity and fatigue damage can be ignored. For proportional loading, the upper bound which is based on limit analysis, is similar in form to that for constant loading. In the more general case, the upper bound is less stringent and is based on the mean load and temperature distribution over the lifetime. A lower bound on life is taken as the time for the first part of the structure to fail.The bounds are applied to three simple structures. For proportional loading the upper bound predicts the lifetime with the same accuracy as for constant loading except for extreme load variations. The presence of a temperature distribution alters the accuracy of the upper bound prediction but in most cases the change is small. In contrast, the lower bound is very sensitive to the temperature gradient.The authors use these results to develop approximate techniques for estimating the creep life of components subjected to variable loads and temperature distributions. Simplified design procedures based on the upper bound are examined and suitable amendments are proposed.     

不排水双层粘性土地基极限承载力的数值分析

采用弹塑性有限元分析了条形基础作用下不排水条件的双层粘性土地基极限承载力性状。采用修正的地基承载力系数表征,并将不同的几何与土层参数条件下的数值解与上下限解和经典的经验解进行比较。表明弹塑性位移有限元法可以很好地求解地基的极限承载力问题,其求解得到的修正地基承载力系数与基于下限原理的有限元解很接近,而上限解高估了地基的极限荷载值,传统的经验解在某些条件下却偏小。     

关于蜂窝芯体面外等效剪切模量的讨论

对于六边形蜂窝芯体,其面内等效参数具有确定的解析式,便于应用;相比之下,对于面外等效剪切模量,现有工作只能给出其上下限,由于没有确定的取值,给工程计算带来了困扰。为克服这一矛盾,本文通过Y型蜂窝胞元,针对薄面板的情况,重新分析了芯材的面外等效剪切模量。针对直壁板与斜壁板厚度为1:1和2:1的情况,给出了近似的弹性力学解答,并由此确定出面外等效剪切模量的上限。本方法所确定的剪切模量的上限与文献给出的剪切模量的下限是相同的,从而使该模量也具有确定的解析表达式,方便了数值计算和分析。试验数据和有限元数值分析均验证了本文结论的正确性。     

Nonempty intersection theorems in topological spaces with applications

In this paper, we establish some new nonempty intersection theorems for generalized L-KKM mappings and prove some new fixed point theorems for set-valued mappings under suitable conditions in topological spaces. As applications, an existence theorem for an equilibrium problem with lower and upper bounds and two existence theorems for a quasi-equilibrium problem with lower and upper bounds are obtained in topological spaces. Our results generalize some known results in the literature.     

极限分析的无搜索数学规划算法

本文研究理想刚塑性介质极限载荷因子的计算方法。根据极限分权理论的上限定理,建立了计算极限载荷因子的一般数学规划有限元格式。针对这种格式的特点,提出了一个求解极限载荷因子的无搜索迭代算法。这个算法中采用逐步识别刚性、塑性分区,不断修正目标函数的方案,克服了目标函数非光滑所导致的困难。本文提出的算法建立于位移模式有限元基础上,有较广的适用范围,且具有计算效率高,稳定性好,格式简单易于程序实现等优点。     

Non-shakedown collapse of a doubly-reinforced rectangular plate under cyclic loads

A typical doubly-reinforced concrete rectangular plate is subjected to quasi-static (and more general quasiperiodic dynamic) transverse loads. The amount of reinforcements can be different in the upper and lower layers, in the central and rear parts of the plate, and in different directions, as usually designed in practice. An upper bound kinematic approach, which involves construction of potential collapse kinematic fields with plastic hinge lines, is developed to evaluate the non-shakedown loads corresponding to the respective collapse modes. The relations between the non-shakedown load parameters (frequency, amplitude limits) and the reinforcement parameters are derived for practical use. The kinematic assumptions with plastic hinge lines reduce the set of admissible kinematic fields for our upper bound approach, however the procedure appears relatively simple, visual, and can be developed to investigate the behaviour of other plates in various loading and reinforcement schemes, like the respective approach of plastic limit analysis, which was restricted to static loading.     

The application of graph theory to the limit analysis for torsion of thin-walled bars with complex cross-sections

Graph theory is employed in this paper as a means to establish the topological model of complex thin-walled cross-sections. On this basis, the upper and lower bound theorems of the plastic limit analysis are applied to the analysis of the plastic limit shear flows on the cross-section of thin-walled bars under St. Venant torsion. Corresponding mathematical programming problems are formulated and their duality is shown. After solving the linear programming problem corresponding to the lower bound theorem, the limit torsional moment of a thin-walled cross-section can be calculated according to the shear stress distribution in the limit state. The formula for calculating the limit torsional moment is given. Furthermore, the limit state of thin-walled cross-sections under St. Venant torsion is also discussed and the concept of the limit tree is introduced. A computer program has been developed by the author. Results calculated by the program for typical complex cross-sections are given.     

On kinematic method in shakedown theory: I. Duality of extremum problems

A kinematic method for determining the safety factor in shakedown problems is developed. An upper bound kinematic functional is defined on a set of kinematically admissible time-independent velocity fields. Every value of the functional is an upper bound for the safety factor. Using convex analysis methods, conditions are established under which the infimum of the kinematic upper bounds equals the safety factor, in particular, conditions under which it is sufficient to consider only smooth velocity fields for the safety factor calculation. The method generalizes that recently proposed for the case of spherical yield surfaces by Kamenjarzh and Weichert. The extension covers a wide class of yield surfaces and inhomogeneous bodies. A shakedown problem for a beam subjected to a concentrated load is considered as an example.     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Spinning rods, elliptical disks and solid ellipsoidal bodies: Elastic and plastic stresses and limit spins

The analysis of the stresses in one-, two- and three-dimensional spinning bodies is discussed in a systematic and comprehensive way. First elastic solutions are derived for rods, for elliptical-shaped flat disks and for ellipsoidal solid bodies spinning about their sideways axes. Then the spins for first plastic yield are found in each case using each of the Tresca and the von Mises yield conditions. Then upper and lower bounds on the maximum allowable limit spins where the body would globally fail assuming perfectly plastic behavior are derived. The elastic solutions at first yield always give a lower bound to that limit spin, but global failure generally does not occur until the spin is increased. A way to calculate an improved lower bound is illustrated. Upper bounds are found in a simple and new way. The method uses the fact that the volume-averaged stresses can be calculated directly from the loadings without the need for any actual stress solutions, and then it is proved that the use of those average stresses in the yield functions always gives an upper bound to the limit loads. That use of the statically determinate average stresses to obtain meaningful plastic upper bounds to limit loads is though to be a new method, and can be applied to any shape. Finally, several finite element calculations are used to determine the quantitative relations between the lower and upper bounds and the actual limit spins for ellipsoidal bodies.The results are of interest in the spin of planetary bodies, where they explain the nature of an average-stress approximate method, and in the analysis of spinning bodies in general. In addition, the approach gives a very interesting example of the utility of the limit analysis approaches of plasticity theories.     

On the equivalence of slip-line fields and work principles for rigid–plastic body in plane strain

Extremum/work principles for a rigid–plastic body have been discussed in classical theory of plasticity to be of immense significance. Unfortunately, till now, these extremum theorems have been used only as a crude method of obtaining the limit load of a rigid–plastic body, using successive approximations by upper and lower bound estimates. On the other hand slip-line fields (SLF) have been extensively used not only for evaluation of limit load but also for obtaining sufficiently accurate estimates of stresses in the plastic region as well as in the vicinity of crack tip. Till now, these two methods of plastic analyses, that is, the work principles and SLF have remained more or less independent apart from the fact that both are upper bounds as they use kinematically admissible velocity fields. Recently, a new load bounding technique, modified upper bound (MUB) Approach, was proposed by Khan and Ghosh [Khan, I.A., Ghosh, A.K., 2007. A modified upper bound approach to limit analysis for plane strain deeply cracked specimens. International Journal of Solids and Structures 44 (10), 3114–3135]. In this article, a rigorous mathematical basis of this load bounding technique is presented and it is demonstrated that the method is actually a new form of the general extremum/work principles. The equivalence of this new form of work principle, that is, MUB with the classical SLF analysis, for a rigid–plastic material in plane strain, has been discussed in detail. Since plastic deformation fields depend on specimen geometry and type of loading specific cases have been considered. Both cracked and uncracked configurations have been analysed to establish this equivalence in general. Various simplifications resulting from the use of this new load bounding technique over SLF method has been demonstrated. Several standard problems of plane strain analysed by SLF method and validated by experiments in past have been considered in this article. As a novel application of the proposed method, single-edge-cracked plate under combined bending and tensile load has been analysed. For this specimen SLF solutions are available only for bending with small tensile load (defined in Section 3.2.4) while classical upper bound solutions are valid for bending with large tensile load. In this work a completely analytical formulation for yield locus for the entire range of tensile and bending load has been obtained. Apart from accurate evaluation of limit load, detailed evaluation of crack tip stresses and hence constraint near the crack tip has been performed using this new form of work principles.