Numerical lower bound shakedown analysis of engineering structures

We propose a numerical method for the computation of shakedown loads of engineering structures subjected to varying thermo-mechanical loading. The method is based on Melan’s lower bound shakedown theorem using the von Mises yield criterion. The resulting nonlinear convex optimization problem is presented in a generalized formulation and then solved by an interior-point algorithm, which is characterized by a problem-tailored solution strategy, particularly suitable for application to large-scale engineering structures.Theoretical and numerical issues of the algorithm are described. It’s efficiency is shown by application to thermo-mechanical problems from power plant engineering. The results are compared to those found in literature as well as to calculations with other optimization codes lancelot, ipopt and ipdca.     

Plastic behaviour and stability constraints in the shakedown analysis and optimal design of trusses

A shakedown analysis and optimum shakedown design of elasto-plastic trusses under multi-parameter static loading are presented. To control the plastic behaviour of the truss, bounds on the complementary strain energy of the residual forces and on the residual displacements are applied and for the bars under compression critical stresses updated during the iteration are taken into consideration. The formulation of problems is suitable for nonlinear mathematical programming which is solved by the use of an iterative procedure. The application of the method is illustrated by three test examples.     

A NLP approach for evaluating storey-buckling strengths of steel frames under variable loading

The problem of determining elastic buckling strengths for unbraced steel frames under variable loading is investigated in this paper. Whereas the pattern of applied loads is specified prior to stability analysis of a frame under proportional loading, load patterns are not predefined in variable loading. The conventional methods for evaluating the stability strength of unbraced frames under proportional loading are not applicable for variable loading, since the load pattern is unknown. Taking into account the concept of storey-based buckling, the problem of frame stability under variable loading is presented as a pair of minimization and maximization problems subject to stability constraints, which are solved by a nonlinear programming (NLP) method. The proposed variable loading approach takes into account the variability of applied loads during the life span of the structure, and as such, provides accurate evaluation of elastic frame-buckling strengths.     

Computational methods for optimal shakedown design of FE structures

The paper concerns the optimal shakedown design of structures discretized by elastic perfectly plastic finite elements. The design problem is formulated in four alternative versions, i.e. as the search for the minimum volume design whose shakedown limit load multiplier is assigned or as the search for the maximum shakedown limit load multiplier design whose volume is assigned; both problems are approached on the grounds of the shakedown lower bound and upper bound theorems. Correspondingly four computational methods, one for each original problem, are presented. These methods consist in solving iteratively new problems which are simpler than the original ones, but expressed in such a way that the obtained design and behavioural variables fulfill the optimality conditions of the relevant original problems, and thus they provide the true optimal design. Finally, an alternative numerical approach devoted to obtaining the optimal shakedown design is presented. Several numerical examples confirm the theoretical results.     

A simplified method of solution for the short cycle creep-plasticity problem

This paper describes a method for estimating the long-term effects on structures under cyclic changes of loading. For loads less than a certain critical amplitude (shakedown limit), the stress in the structure will a symptote to a cyclic stationary state consisting of an elastic part in response to the cyclic loading, plus a system of self-equilibrating residuals constant in time. It is shown that corresponding to this cyclic stationary state, the creep energy dissipation per cycle of loading is a maximum. Instead of following the exact time history to reach this state, in this paper it is found by a procedure of successive approximations. It corrects the admissible residual stress distribution at the beginning of a cycle by the creep and plastic strains accumulated over an entire cycle, which are in general not compatible, and requires additional self-equilibrating stresses to give an elastic strain distribution such that the total strain satisfies compatibility. The steady state is reached when no further correction is necessary. Convergence may be accelerated by a suitable choice of initial starting value, and by an artificial choice of the cycle time for the best computational convenience, upon which the steady-state solution can be proved to be independent. The procedure is a powerful device to obtain the cyclic steady-state solution, which will give an upper bound to the creep deformation per cycle and may also be used to find the shakedown limit. The formulation of the procedure in conjunction with the finite element method is given in detail and results of a few examples of the analysis are shown.     

Optimal shakedown design of beam structures

The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such conditions prove the equivalence of the two types of design problems, provide useful information on the structural behaviour in optimality conditions, and constitute a fifth possible way to determine the optimal design. Whatever approach is used, the strong non-linearity of the corresponding problem does not allow the finding of the analytical solution. Consequently, in the application stage suitable numerical procedures must be employed. Two numerical examples are given.     

基于ANSYS APDL语言实现复杂载荷加载

针对通用有限元分析软件现有载荷形式无法进行复杂载荷加载的问题,以某船用柴油机惯性载荷连杆的静力分析为例,提出基于ANSYS APDL进行复杂载荷加载的方法. 对该连杆在复杂载荷条件下的零部件力学仿真分析表明,此方法可方便、高效地实现不同节点惯性力的计算,完成众多节点力的加载.     

Optimal shakedown design of frames by linear programming

A mathematical model of flexural framed structures subjected to cyclic loadings is presented, and the problem of minimum-weight shakedown design of these structures is formulated as a pair of dual linear programming problems. Such a formulation reflects the duality between static and kinematic theorems of shakedown theory. An optimal solution for plastic moment distribution realizes a minimum volume of a structure and ensures a given safety factor against plastic collapse by inadaptation to a prescribed cyclic loading program. The reduced-size form of the problem is presented where the bending moments are expressed in terms of independent redundants. This form is more attractive from the computational point of view for the use of linear programming codes. Several examples illustrate the presented method.     

Nonlinear dynamic analysis of frame structures

A geometrically nonlinear dynamic analysis method is presented for frames which may be subjected to finite rotations in three-dimensional space. The proposed method is based on the static geometrically nonlinear analysis method reported by Yoshida et al., in which the governing incremental equilibrium equation is represented by the coordinates after the deformation themselves rather than conventional displacements. The governing dynamic equilibrium equation for each element is obtained from the static equation by adding the inertia term. In the solution procedure, a modified Steffensen's iteration process is introduced and combined with the two-step approximation and iterative correction solution procedure developed for static analysis. A numerical example of a curved cantilever beam under lateral loads indicates the effectiveness of the proposed method in cases with three-dimensional finite rotations. Forced vibration analyses of a two-hinged shallow arch are conducted under centrally concentrated loading with several loading amplitudes. The resulting dynamic buckling load is compared with that given by Gregory and Plaut in 1982, who used Galerkin method, and shows good agreement.     

Optimal shakedown loading for circular plates

Optimization of shakedown loading under constrained residual displacement is considered for elastic and perfectly plastic circular plates. The load variation bounds, which satisfy the optimality criterion in concert with plate-strength and stiffness requirements, are identified. The actual strain fields of the plate depend on the loading history. Thus, the load optimization problem at shakedown is stated as a pair of problems that are executed in parallel: the main load optimization and the verification of the prescribed magnitudes of the bounds on the residual deflections. The problem must be solved by iteration. The Rozen projected gradient method is applied. A mechanical interpretation of the Rozen optimality criterion is given, which permits the simplification of the mathematical model for load optimization and the formulation of the solution algorithm. Numerical examples include circular annular plates with and without a rigid inclusion. The results are valid under the assumption of small displacements.     

Optimization of flexible components of multibody systems via equivalent static loads

An optimization methodology that iteratively links the results of multibody dynamics and structural analysis software to an optimization method is presented to design flexible multibody systems under dynamic loading conditions. In particular, rigid multibody dynamic analysis is utilized to calculate dynamic loads of a multibody system and a structural optimization algorithm using equivalent static loads transformed from the dynamic loads are used to design the flexible components in the multibody dynamic system. The equivalent static loads, which are derived from equations of motion, are used as multiple loading conditions of linear structural optimization. A simple example is solved to verify the proposed methodology and the pelvis part of the biped humanoid, a complex multibody system which consists of many bodies and joints, is redesigned using the proposed methodology.     

Shape optimization of structures under earthquake loadings

The optimum design of structures under static loads is well-known in the design world; however, structural optimization under dynamic loading faces many challenges in real applications. Issues such as the time-dependent behavior of constraints, changing the design space in the time domain, and the cost of sensitivities could be mentioned. Therefore, optimum design under dynamic loadings is a challenging task. In order to perform efficient structural shape optimization under earthquake loadings, the finite element-based approximation method for the transformation of earthquake loading into the equivalent static loads (ESLs) is proposed. The loads calculated using this method are more accurate and reliable than those obtained using the building regulations. The shape optimization of the structures is then carried out using the obtained ESLs. The proposed methodologies are transformed into user-friendly computer code, and their capabilities are demonstrated using numerical examples.     

Structural dynamic topology optimization based on dynamic reliability using equivalent static loads

An approach for reliability-based topology optimization of interval parameters structures under dynamic loads is proposed. We modify the equivalent static loads method for non linear static response structural optimization (ESLSO) to solve the dynamic reliability optimization problem. In our modified ESLSO, the equivalent static loads (ESLs) are redefined to consider the uncertainties. The new ESLs including all the uncertainties from geometric dimensions, material properties and loading conditions generate the same interval response field as dynamic loads. Based on the definition of the interval non-probabilistic reliability index, we construct the static reliability topology optimization model using ESLs. The method of moving asymptotes (MMA) is employed as the optimization problem solver. The applicability and validity of the proposed model and numerical techniques are demonstrated with three numerical examples.     

Optimum shakedown design under residual displacement constraints

We consider the minimum weight design of suitably discretized elastoplastic structures subjected to variable repeated loads and constraints on the amount of residual (or permanent) deflections. The optimization problem is formulated on the basis of the classical lower bound theorem of shakedown, supplemented by appropriate constraints on deflections obtained from existing bounding results. The primary purpose of the paper is to show that, even for large size structures, this important and challenging problem can be modeled and solved directly as a mathematical programming problem within the industry standard modeling framework GAMS. Examples concerning truss-like structures are presented for illustrative purposes. Received January 18, 1999     

Topology optimization with design-dependent pressure loading

In this paper, the layout of structures under design-dependent pressure loading is optimized using a topology optimization approach. In contrast to topology optimization problems with conventional static external loading, the position and direction of pressure loading are changing with topology of structure during optimization iterations. In order to model the changing structural surface boundaries under design-dependent pressure loading, a pseudo equal-potential function is introduced. Design sensitivity analysis is derived from the adjoint method. Three examples solved by the proposed method are presented.     

Nonproportional loading limit for structures

The design of most structures involves insuring integrity for several independent loading conditions. Often, it is rational to design these exploiting the ductility of the materials to reduce structural costs. Then, insuring integrity evokes the need for non-monotonic limit analyses.This paper lays the groundwork for implementing these analyses. It provides an alternate limit load characterization to that of Greenberg, formulates a mathematical statement of the problem leading to both monotonie (proportional) and non-monotonic limit loads values, describes a direct limit toad analysis procedure for analysis within a finite element framework, and evaluates limit loads for sets of simple structures.This paper describes a direct limit analysis process and characterizes non-monotonic limit loads. The process furnishes “exact” values of limit loads with increasing efficiency as the number of structural elements and force redundancy decreases. Tests show the accuracy of predicting non-monotonic limit loads is very sensitive to the number of degrees of freedom in the analysis compared with the total number in the model.Problem solutions show that the magnitude of the non-monotonic limit loading may be greater or less than the load at which yielding first occurs, depending on the structural configuration. Structures designed to not yield may fail at lower loads than the yield loading after non-monotonic plastic loading occurs. Limit loads may increase or decrease as redundancy increases. Although all structures have a proportional loading limit, structures can be designed to preclude yielding in their unloaded state after limit loading is removed.The paper concludes that economy of material cannot be guaranteed by designing for limit load integrity as opposed to designing to prevent the occurrence of yield. The advantages of exploiting the ductility of the material exist only when the structural form is suitable.     

Conditions for structural optimality

Typical difficulties encountered in the formulation of problems of optimal structural design and the inadequacy of widely used empirical optimality criteria are illustrated by examples. For the optimal design of a statically determinate or indeterminate truss of given layout, a method is presented by which necessary and sufficient conditions for global optimality may be derived when an upper bound is prescribed for the compliance of the truss under one or several sets of loads and a lower bound is prescribed for the cross sectional area of each bar. The extension of the method to other structures and constraints is briefly discussed with reference to the literature, and the general form of the resulting optimality conditions is given.     

A linear programming approach to shakedown analysis of structures

Two and three dimensional structures are dealt with, subjected to variable repeated loads, in order to establish a numerical tool for determining the load domain multiplier that gives rise to shakedown. The structure is made discrete by finite elements and the yield domain is linearized. By applying Bleich and Melan's theorem, two primal static formulations are found in linear programming, from which the relevant dual kinematic versions are obtained via duality properties.Numerical results are given at the end of the paper, together with some considerations about the numerical efficiency of the proposed formulations.     

Finite-element dynamic inelastic analysis of beams and plates under combined loading

The response of beams and plates under multiple loading is considered taking into account the inelastic interactions between moments and axial (in-plane) forces. The plastic analysis is based on distribution of stresses and strains in three dimensions so that yielding remains a function of the uniaxial state of stress in beams and of the biaxial state of stress in plates. The interaction effects between dynamic transverse loads and static in-plane loads is studied by using initial stress stiffness matrices that modify the original stiffness matrices. Detailed response behavior of beams and plates in combined loading is presented. The response characteristics are found to exhibit interaction instability properties.     

Optimal design of elastic plastic frames accounting for seismic protection devices

The optimal design of elastic perfectly plastic steel frames with or without suitable protection devices and subjected to static as well as seismic loadings is studied. Two minimum volume problem formulations are proposed, on the grounds of the so-called statical approach, accounting for three different resistance limits: the purely elastic limit, the (elastic) shakedown limit and the instantaneous collapse limit. The adopted load combinations are characterized by the presence of fixed loads, of quasi-static perfect cyclic loads and dynamic (seismic) loads. The linear elastic effects of the dynamic actions are studied by utilizing a modal technique. The proposed treatment is referred to the most recent Italian code related to the structural analysis and design. The solution of the optimization problem is reached by using an appropriate linearization iterative technique specialized to the proposed formulations. Flexural frames and cross-braced frames are studied, and the related minimum volume structures are reached for assigned features of the base isolation device. The Bree diagrams of the obtained optimal designs are also determined in order to characterize their structural behaviour.