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, some 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 state 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 be 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 stress and plastic strain fields, the shakedown analysis of the structure can be reduced to a mathematical programming problem. The results of calculation show that the shakedown load of 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 a scientific basis to improve the design of shell structures.     

A comparison between analytical calculations of the shakedown load by the bipotential approach and step-by-step computations for elastoplastic materials with nonlinear kinematic hardening

The class of generalized standard materials is not relevant to model the nonassociative constitutive equations. The bipotential approach, based on a possible generalization of Fenchel’s inequality, allows the recovery of the flow rule normality in a weak form of an implicit relation. This defines the class of implicit standard materials. For such behaviours, this leads to a weak extension of the classical bound theorems of the shakedown analysis. In the present paper, we recall the relevant features of this theory. Considering an elastoplastic material with nonlinear kinematic hardening rule, we apply it to the problem of a sample in plane strain conditions under constant traction and alternating torsion in order to determine analytically the interaction curve bounding the shakedown domain. The aim of the paper is to prove the exactness of the solution for this example by comparing it to step-by-step computations of the elastoplastic response of the body under repeated cyclic loads of increasing level. A reliable criterion to stop the computations is proposed. The analytical and numerical solutions are compared and found to be closed one of each other. Moreover, the method allows uncovering an additional ‘2 cycle shakedown curve’ that could be useful for the shakedown design of structure.     

Shakedown analysis of engineering structures with limited kinematical hardening

We present a numerical method for the computation of shakedown loads of engineering structures with limited kinematical hardening under thermo-mechanical loading. The method is based on Melan’s statical shakedown theorem, which results in a nonlinear convex optimization problem. This is solved by an interior-point algorithm recently developed by the authors, specially designed for lower bound shakedown analysis of large-scale problems. Limited kinematical hardening is taken into account by use of a two-surface model, such that both alternating plasticity and incremental collapse can be captured. For the yield surface as well as for the bounding surface the von Mises criterion is used. The proposed method is validated by two examples, where numerical results are compared to those of literature where available.     

Relationships of LA theorems for NRT structures by means of duality

The paper refers to previous works developed by the authors, dealing with the possibility of applying duality theorems to non-linear programs coming out from limit analysis (LA) of structures made by not resisting tension (NRT) or no-tension material.Under such perspective, after setting up the static or kinematic LA problem for NRT structures, the main task, for duality theorems to be applicable, is to demonstrate some convexity-related properties of the involved functions and domains.This feature, which is of basic importance for the whole procedure, is not trivial since all of the required conditions are to be accurately checked by analytical developments.Application of duality is finally demonstrated to give a complete and clear interpretation from a physical point of view about relationships relevant to LA approaches.     

On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization

The thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening is analyzed within the present paper. This coupling is of utmost importance in many applications, e.g., in those showing low cycle fatigue (LCF) under large strain amplitudes. Since the by now classical thermomechanical coupling originally proposed by Taylor and Quinney cannot be used directly in case of kinematic hardening, the change in heat as a result of plastic deformation is computed by applying the first law of thermodynamics. Based on this balance law, together with a finite strain plasticity model, a novel variationally consistent method is elaborated. Within this method and following Stainier and Ortiz (2010), all unknown variables are jointly and conveniently computed by minimizing an incrementally defined potential. In sharp contrast to previously published works, the evolution equations are a priori enforced by employing a suitable parameterization of the flow rule and the evolution equations. The advantages of this parameterization are, at least, twofold. First, it leads eventually to an unconstrained stationarity problem which can be directly applied to any yield function being positively homogeneous of degree one, i.e., the approach shows a broad range of application. Secondly, the parameterization provides enough flexibility even for a broad range of non-associative models such as kinematic hardening of Armstrong–Frederick-type. Different to Stainier and Ortiz (2010), the continuous variational problem is approximated by a standard, fully-implicit time integration. The applicability of the resulting numerical implementation is finally demonstrated by analyzing the thermodynamically coupled response for a loading cycle.     

Determination of corrective local compliances of structures for use in vibration computations

Usually, thin-walled structures, in particular, aircraft, launchers, and space structures, are irregular compound systems. In structure dynamics, they are usually represented as separate mutually interacting substructures [1–3]. Very often, different substructures are connected at separate nodes. This is typical of separable blocks, suspensions, and transformable details [4]. At such nodes, considerable local strains and compliances may arise in connected substructures, which significantly affects the dynamic characteristics of the entire structure. In engineering practice, the local compliances of a structure at connection nodes are often modeled by equivalent springs whose characteristics are determined by approximate methods [5] and expeirents.The substructure synthesis method, the methods for calculating the dynamic characteristics of compound structures, and the methods for their refinement have been considered in numerous papers; here we note the publications [6–10].In the present paper, we propose a method for calculating additional (corrective) local compliances at the connection nodes of compound structures on the basis of quasistatic strain equations and for taking these compliances into account in the equations of dynamics of compound structures obtained by expanding the displacements with respect to lower vibration modes of the substructures.     

Simulation of the aerodynamics of buildings and structures by means of the closed vortex loop method

Problems of the numerical simulation of the air flow past buildings and structures are considered using the closed vortex loop method. A mathematical model, based on the vortex approach, of the time-dependent ideal incompressible fluid flow past a system of bodies is proposed. A numerical scheme for solving the problem and an algorithm for calculating the distributed wind loads over the body surface are outlined. An example of calculating the aerodynamic loads is given for a real building and the results are compared with the known results of testing a model of the building in a wind tunnel. An example of the calculation and analysis of the wind distribution over a system of several buildings is also presented.     

Towards real-time simulation of deformable structures by means of co-rotational finite element formulation

Meccanica - Whereas typical Finite Element (FE) computations are performed off-line, many virtual-reality (VR) applications put a demand for interactive simulations involving deformable objects....     

Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures

Shakedown analysis is an extension of plastic limit analysis to the case of variable repeated loads and plays a significant role in safety assessment and structural design. This paper presents a solution procedure based on the meshless local Petrov–Galerkin (MLPG) method for lower-bound shakedown analysis of bounded kinematic hardening structures. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the targeted domain. Moreover, the natural neighbour interpolation (NNI) is employed to construct trial functions for simplifying the imposition of essential boundary conditions. The kinematic hardening behaviour is simulated by an overlay model and the numerical difficulties caused by the time parameter are overcome by introducing the conception of load corner. The reduced-basis technique is applied to solve the mathematical programming iteratively through a sequence of reduced residual stress subspaces with very low dimensions and the resulting non-linear programming sub-problems are solved via the Complex method. Numerical examples demonstrate that the proposed solution procedure is feasible and effective to determine the shakedown loads of bounded kinematic hardening structures as well as unbounded kinematic hardening structures.     

Tracking of coherent thermal structures on a heated wall by means of infrared thermography

This paper deals with measurements of convective velocity of large-scale thermal structures, using the thin foil technique and infrared thermography to visualize the thermal pattern on the wall. An image correlation method is proposed to track the displacement of the observed thermal pattern. The idea of the method is similar to that of particle image velocimetry, but the thermal patterns on the heated wall are used, rather than tracing particles. On this basis, the thermal patterns created by the coherent structures of turbulent channel flow are examined. Particular attention is paid to the determination of the optimal parameters of image acquisition, including spatial and temporal separation. An attempt is made to relate momentum and scalar transport analyses by considering the propagation velocity of large-scale temperature structures. The proposed technique appears to be an attractive alternative for non-intrusive analysis of turbulent flow, especially, where opaqueness of channel walls excludes the use of optical methods. Received: 18 January 2000/Accepted: 20 May 2000     

Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

The stationary response of Duffing oscillator with hardening stiffness and fractional derivative under Gaussian white noise excitation is studied. First, the term associated with fractional derivative is separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique, and the original system is replaced by an equivalent nonlinear stochastic system without fractional derivative. Then, the stochastic averaging method of energy envelope is applied to the equivalent nonlinear stochastic system to yield the averaged Itô equation of energy envelope, from which the corresponding Fokker–Planck–Kolmogorov (FPK) equation is established and solved to obtain the stationary probability densities of the energy envelope and the amplitude envelope. The accuracy of the analytical results is validated by those from the Monte Carlo simulation of original system.     

梁柱简化模型在自立塔塔线体系动态响应中的适用性研究

对输电塔进行合理简化可以提高塔线体系动力学仿真的效率。本文给出自立塔梁柱简化模型的计算方法,并提出利用梁柱简化模型计算方法建立自立塔塔线体系整体模型,同时采用桁梁混合模型建立精细化塔线体系整体模型,对两种模型塔线体系静力特性及振型和固有频率等动力特性进行对比分析。以脱冰工况为例,采用生死单元技术将施加在输电线节点上的集中质量单元杀死来模拟脱冰,实现对塔线体系动力学响应的有限元模拟,研究塔线体系简化模型在动态响应中的适用性。结果表明,两种模型弯曲变形误差小,低阶的振型相同,固有频率值误差小,动力特性基本相同;脱冰工况下,自立塔节点位移和塔材内力时程曲线一致,在提高计算效率的情况下,能有效保证计算精度。     

Dynamic response of underground structures by time domain SBEM and SFEM

The dynamic interaction problems of three-dimensional linear elastic structures witharbitrary shaped section embedded in a homogeneous,isotropic and linear elastic half spaceunder dynamic disturbances are numerically solved.The numerical method employed is acombination of the time domain semi-analytical boundary element method(SBEM)usedfor the semi-infinite soil medium and the semi-analytical finite element method(SFEM)used for the three-dimensional structure.The two methods are combined throughequilibrium and compatibility conditions at the soil-structure interface.Displacements,velocities,accelerations and interaction forces at the interface between undergroundstructure and soil medium produced by the diffraction of wave by an underground structurefor every time step are obtained.In dynamic soil-structure interaction problems,it isadvantageous to combine the SBEM and the SFEM in an effort to produce an optimumnumerical hybrid scheme which is characterized by the main advantages of the two methods.The     

Stochastic seismic response of long-span structures

In this paper, dynamic responses of long-span structures subjected to the action of earthquake with realistic wave speed are analysed. The horizontal or vertical ground motion due to earthquakes is assumed to be a stationary stochastic process, and the seismic waves travel along a horizontal straight line. Expressions for calculating the psd(power spectral density) matrices of structural displacements and internal forces are derived based on three dimensional FEM structural models with the ground motion phase-lags taken into account. A numerical example is given which shows that it is of great importance to consider the effect of the ground motion phase-lags for long-span structures.This work was supported by the National Natural Science Foundation of China.     

Compressive response and failure of braided textile composites: Part 2—computations

In this, the second part of a two part paper, results obtained by using the finite element (FE) method in conjunction with micromechanics to predict the effective elastic stiffness and strength of a carbon 2D triaxially braided composite (2DTBC), are presented. The 3D FE based micromechanics study was carried out on one representative unit cell (RUC) of the carbon 2DTBC (the “micromodel”). The FE models were first used to determine the macroscopic elastic orthotropic stiffnesses of the 2DTBC. The micromodel was deemed acceptable (in terms of the number of elements used in the mesh of the micromodel) if the elastic stiffnesses it displayed were within 5% of the elastic properties found experimentally. Subsequently, buckling eigenmodes were determined for the FE RUC under uniaxial and biaxial loading states, corresponding to the experimental investigation reported in part I of this two part paper. The lowest symmetric modes were identified and these mode shapes were used as imperfections to the FE model for a subsequent nonlinear response analysis using an arc-length method in conjunction with the ABAQUS commercial FE code. The magnitude of the imperfections was left as a parameter and its effect on the predicted response was quantified. The present micromechanics computational model provides a means to assess the compressive and compressive/tensile biaxial strength of the braided composites and its dependence on various microstructural parameters. It also serves as a tool to assess the most significant parameter that affects compressive strength.