Optimization of rigid-plastic shallow spherical shells of piecewise constant thickness

An approximate method developed earlier for the investigation of large plastic deflections of circular and annular plates is accommodated for shallow spherical shells. The material of the shells is assumed to obey Tresca's yield condition and the associated deformation law. The minimum weight problem concerning shells operating in the post-yield range is posed under the conditions that (i) the thickness of the structure is piece-wise constant and (ii) the maximal deflections of the optimized shell and a reference shell of constant thickness, respectively, coincide. Necessary optimality conditions are derived with the aid of the variational methods of the optimal control theory. The set of equations obtained is solved numerically.     

Optimization of plastic spherical shells with a central hole

An optimization method for plastic spherical shells is presented. The shells under consideration are clamped at the outer edge and contain a central hole. The material of the shells obeys the generalized square yield condition and the associated flow rule. The problem of maximization of the load carrying capacity under the condition that the weight (material volume) of the shell is fixed is transformed into a problem of nonlinear programming. The latter is solved with the aid of Lagrangian multipliers. The solution obtained is compared with the optimal solution of the minimum weight problem for given load carrying capacity. Received September 9, 1999     

Integrated optimization of the material and structure of composites based on the Heaviside penalization of discrete material model

Based on discrete material optimization and topology optimization technologies, this paper discusses the problem of integrated optimization design of the material and structure of fiber-reinforced composites by considering the characteristics of the discrete variable of fiber ply angle because of the manufacture requirements. An optimization model based on the minimum structural compliance with a specified composite volume constraint is established. The ply angle and the distribution of the composite material are introduced as independent variables in two geometric scales (material and structural scales). The void material is added into the optional discrete material set to realize the topology change of the structure. This paper proposes an improved HPDMO (Heaviside Penalization of Discrete Material Optimization) model to obtain a better convergent result, and an explicit sensitivity analysis is performed. The effects of the HPDMO model on the convergence rate of the optimization results, the objective function value and the iteration history are studied and compared with those from the classical Discrete Material Optimization model and the Continuous Discrete Material Optimization model in this paper. Numerical examples in this paper show that the HPDMO model can effectively achieve the integrated optimization of the fiber ply angle and its distribution in the structural domain, and can also considerably improve the convergence rate of the optimal results compared with other DMO models. This model will help to reduce the manufacture cost of the optimal design.     

Optimization of conical shells of Mises material

Conical shells made of a von Mises material are considered. The shells are subjected to unifromly distributed lateral loading and are simply supported at outer edges whereas inner edges are absolutely free. The shell wall is assumed to be of piece-wise constant thickness. Resorting to the lower bound theorem of limit analysis, optimal designs of shells are established under given weight (material volume of a shell) which corresponds to the maximum load carrying capacity. Received May 15, 2000     

Optimization of plastic spherical shells of von Mises material

An optimization procedure is developed for spherical shells pierced with a central hole. The outer edge of the shell is simply supported whereas the inner edge is absolutely free. The material of the shell is assumed to be an ideal plastic material obeying the von Mises yield condition. Resorting to the lower bound theorem of limit analysis, shells with constant and piece-wise constant thickness are considered. The designs of spherical shells corresponding to maximal load carrying capacity are established for a given weight. Necessary optimality conditions are derived with the aid of variational methods from the theory of optimal control. The obtained set of equations is solved numerically.     

Optimal design of plastic cylindrical shells of von Mises material

Optimal design of plastic circular cylindrical shells of von Mises material is studied. The optimization problem is stated as the maximization problem of the load carrying capacity for given weight of the shell. Shells with constant and piecewise-constant thickness are considered. The maximization problem is performed under the requirement that the material volume of the stepped shell is equal to the case of the reference shell of constant thickness. The material of the shell is assumed to be an ideal rigid plastic obeying von Mises yield criterion. The considered nonlinear problems are solved by using the CASes method.     

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.     

Postbuckling behaviour of plates and shells using a mindlin shallow shell formulation

The paper presents a finite element Mindlin shallow shell formulation. Compared to a previous flat plate formulation it is shown that the addition of a shallow shell capability adds very little extra computational effort. Results are given for the postbuckling behaviour of square and circular plates subject to direct inplane loading and a square plate subject to inplane shear loading. Examples are also presented of the analyses of a shallow truss and cylindrical and spherical shells, all exhibiting snap through behaviour. Agreement with existing solutions is generally good and where possible the results are presented numerically.     

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.     

Optimal design of structural steel frameworks

This paper reviews work conducted at the University of Waterloo during the 1980s concerning the computer-automated design of least-weight structural steel frameworks. First, design under static loads is considered whereby the members of the structure are automatically sized using commercial steel sections in full conformance with design standard provisions for elastic strength/stability and stiffness. This problem is illustrated for the least-weight design of a steel mill crane framework comprised of a variety of member types and subject to a number of load effects. Then, the design methodology is extended to the least-weight design of structural steel frameworks under both service and ultimate loading conditions. Here, acceptable elastic stresses and displacements are ensured at the service-load level while, simultaneously, adequate safety against plastic collapse is ensured at the ultimate-load level. This design problem is illustrated for the least-weight design of an industrial steel mill framework for which plastic behaviour is governed by conservative piecewise linear yield conditions. Finally, the computer-based design methodology is extended to the least-weight design of structural steel frameworks subjected to dynamic loading. Constraints are placed on dynamic displacements, dynamic stresses, natural frequencies and member sizes. The design problem is illustrated for the least-weight design of a steel trussed arch subjected to non-structural masses and an impulse force.     

Elasto-plastic contact stress analysis of joints subjected to cyclic loading

Rigorous elastic-plastic finite element analysis of joints subjected to cyclic loading is carried out. An incremental-iterative algorithm is developed in a modular form combining elasto-plastic material behaviour and contact stress analysis. For the case of the interference fit, the analysis sequentially carries out insertion of the pin and application of the load on the joint, covering possible initiation of separation (and/or yielding) and progressively the receding/advancing contact at the pin-plate interface. Deformations of both the plate and the pin are considered in the analysis. Numerical examples are presented for the case of an interference fit pin in a large plate under remote cyclic tension, and for an interference fit pin lug joint subjected to cyclic loading. A detailed study is carried out for the latter problem considering the effect of change in contact/separation at the pin-plate interface on local stresses, strains and redistribution of these stresses with the spread of a plastic zone. The results of the study are a useful input for the estimation of the fatigue life of joints.     

Optimization of plastic shallow shells exhibiting nonstable behaviour

An optimal design technique is suggested for axisymmetric shallow shells exhibiting nonstable behaviour in the post-yield point range. The material of the shells is ideally rigidplastic obeying the von Mises yield condition, which is satisfied in the average and the associated deformation law. Spherical shells pierced with a hole and subjected to uniformly distributed transverse pressure are studied. Different cases of the support conditions are considered. Making use of the methods of the optimal control theory the problem is transformed into a boundary value problem which is solved numerically.     

On post-buckling analysis and experimental correlation of cylindrical composite shells with Reissner-Mindlin-Von Kármán type facet model

In this paper post-buckling analysis of carbon fibre reinforced plastic cylindrical shells under axial compression is considered. Reissner-Mindlin-Von Kármán type shell facet model is used in the computations. The effect of geometric imperfection shape and amplitude on nonlinear analysis results is discussed. Numerical-experimental correlation is performed using the results of experimental buckling tests found in the literature. Results show that bringing the diamond shape geometric imperfection in the model significantly improves the correlation and gives good accuracy in simulating cylindrical shell post-buckling behaviour.     

Optimization of shallow shells with different yield stresses in tension and compression

The minimum weight problem of thin rigid-plastic shallow spherical shells is studied. The thickness of the shell is piece-wise constant and the material has different yield stresses in tension and compression. The flow theory of plasticity is employed. Both solid and sandwich shells are considered. Necessary optimality conditions are derived with the aid of optimal control theory.     

Multi-fidelity optimization of laminated conical shells for buckling

Optimum laminate configuration for minimum weight of filament-wound laminated conical shells is investigated subject to a buckling load constraint. In the case of a composite laminated conical shell, due to the manufacturing process, the thickness and the ply orientation are functions of the shell coordinates, which ultimately results in coordinate dependence of the stiffness matrices (A,B,D). These effects influence both the buckling load and the weight of the structure and complicate the optimization problem considerably. High computational cost is involved in calculating the buckling load by means of a high-fidelity analysis, e.g. using the computer code STAGS-A. In order to simplify the optimization procedure, a low-fidelity model based on the assumption of constant material properties throughout the shell is adopted, and buckling loads are calculated by means of a low-fidelity analysis, e.g. using the computer code BOCS. This work proposes combining the high-fidelity analysis model (based on exact material properties) with the low-fidelity model (based on nominal material properties) by using correction response surfaces, which approximate the discrepancy between buckling loads determined from different fidelity analyses. The results indicate that the proposed multi-fidelity approaches using correction response surfaces can be used to improve the computational efficiency of structural optimization problems.     

Minimum weight design for toroidal pressure vessels using Differential Evolution and Particle Swarm Optimization

In this paper, Differential Evolution and Particle Swarm Optimization methods have been applied to the design of minimum weight toroidal shells subject to internal pressure. Constraints are first yield pressure, plastic pressures, plastic instability pressure and volume contained by the toroid. Optimality includes geometry and wall thickness, which is constant or variable. The optimization process is performed by FORTRAN routines coupled with finite element analysis code ABAQUS. Depending on geometry parameters of the toroid, the material saving can be as high as 72%. The results also show that Differential Evolution outperforms the Particle Swarm Optimization in most of cases.     

作大范围运动弹塑性平面板的动力学分析

研究作大范围运动弹塑性平面板的动力学特性.考虑了几何非线性和材料非线性,基于平面应力假设、Mises屈服条件和流动法则,采用绝对节点坐标法,用虚功原理建立了作大范围运动弹塑性平面板的动力学方程.在数值计算时将各时刻的塑性应变储存在全局数组中,实现了塑性应变的迭代计算.通过对带集中质量、作大范围运动平面板的数值仿真研究塑性效应对系统的动力学特性的影响.     

Integrated structure-electromagnetic optimization of large reflector antenna systems

A novel multiparameter optimization method is developed for use with terrestrial and space reflector antenna electromechanical systems and other metallic and composite engineering structures. To satisfy extremely high design requirements, the proposed approach incorporates the objectives from various structural and electromagnetic (EM) performances of the system at many working/loading cases simultaneously. A finite element method is used for structural analysis. Optical ray tracing, spline function aperture field interpolation, geometric optics aperture integration, and FFT techniques are employed to analyse the EM performances of distorted reflector antennas. A systematic method is used for parameter profile analysis of the system. The optimization involves member size, structural geometric and material design variables. Various terrestrial and orbital working environments and loading cases which affect antenna performances can be included in the optimization model. The optimization of an 8 m antenna system, as an example, is discussed and the results are given.     

Reliability based design with a degradation model of laminated composite structures

Uncertainties in deviations of physical properties lead to a probabilistic failure analysis of the composite materials. The proposed optimization model for laminate composites is based on reliability analysis considering the ultimate failure state. To avoid difficulties associated with the complete analysis of the failure modes, bounds are established for the failure probability of the structural system. These bounds are related with theintact and degraded configurations of the structure. Using thefirst ply failure and thelast ply failure theories and a degradation model for the mechanical properties with load sharing rules we obtain the failure probabilities corresponding to the two above configurations. The failure probability of each configuration is obtained using level 2 reliability analysis and the Lind-Hasofer method.The optimization algorithm is developed based on the problem decomposition into three subproblems having as objectives the maximization of the structural efficiency atintact and degraded configurations of the structure and weight minimization subjected to allowable values for the structural reliability. Additionally, the search for the initial design is performed introducing a weight minimization level. It is expected to explore the remaining load capacity of the structures afterfirst ply failure as a function of the anisotropic properties of the composites. The design variables are the ply angles and the thicknesses of the laminates. The structural analysis for the model developed is performed through the finite element method mainly using the isoparametric degenerated shell finite element. The sensitivities are obtained using the discrete approach through the adjoint variable method. In order to show the performance of the analysis two examples are presented.     

Minimum weight design of symmetric angle-ply laminates under multiple uncertain loads

An angle-ply laminated plate is optimized with the objective of minimizing the weight of the plate taking into account uncertainties in the multiple transverse loads. The weight is proportional to the laminate thickness which is minimized subject to deflection and buckling constraints under the least favourable loading with the ply angles taken as design variables. The convex modelling approach is employed to analyse the uncertain loading with the uncertain quantities allowed to vary arbitrarily around their average values subject to the requirements that these variations are bounded inL ₂ norm and represented by a finite number of eigenmodes. The effect of uncertainty on the optimal design is investigated quantitatively. It is shown that the minimum weight increases with increasing level of uncertainty and the optimal ply angles also depend on the level of uncertainty.