Shape optimization of dynamically loaded viscoelastic cylindrical shells

Axisymmetric deflections of cylindrical shells of variable thickness are examined. The shell material is linear viscoelastic. The loading is of the impulsive type—it induces inside the shell a radial velocity field. The amount of kinetic energy is prescribed. The thickness function includes some design parameters, which must be calculated so that deflections of the beam are minimal. Only designs with a given volume are considered.For solving this optimization problem the space variable and the time will be separated. For evaluating the minimum of the objective function the Nelder-Mead technique has been used. Computations show that the viscosity effect is essential only for very short shells. Some numerical examples are presented.     

Optimum weight design of sandwich cylinders with orthotropic facings and core under combined loads

The minimum weight of a honeycomb sandwich cylinder with the facings in composite material is obtained by an optimization method with the ply angles and the thicknesses of the ply and honeycomb as the design variables. The cylinder is under the combined loads of axial compression, bending moment and transverse shear. The structural analyses verify the stability and the material strength. The stability analyses are done with the simply supported boundary conditions for the orthotropic case. Three stability modes have been studied: general buckling, dimpling and wrinkling. In all cases, the initial imperfections and the prebuckling distortions are not taken into account. This problem is treated with the knock-down factors. For the optimization, we have used a variable metric method for constrained optimization (subroutine VMCON, method of M.J.D. Powell). From an arbitrary design point (feasible or non-feasible), the program searches for an optimal design by the iterative procedure. To verify that the obtained optima are not local, the program has been initiated from several different starting points. Having compared the results, global alternative optima as well as local optima have been found.     

Multicriteria optimization of technological processes under uncertainty conditions

The multicriteria optimization problem under uncertainty conditions is considered, which arises at the design stage of a technological process. The uncertainty stems from the inaccuracy of mathematical models and a certain uncertainty of the conditions in which the process will be implemented. The generalization is treated of well known multicriteria optimization methods in the case of the uncertainty accounting. Here, account will be taken of the possibility of the use of control variables at the operation stage to compensate for the uncertainty effect.     

Non-linear analysis of shells of revolution under arbitrary loads

A new finite element formulation is presented for the non-linear analysis of elastic doubly curved segmented and branched shells of revolution subject to arbitrary loads. The circumferential variations of all quantities are described by truncated Fourier series with an appropriate number of harmonic terms. A coupled harmonics approach is employed, in which coupling between different harmonics is dealt with directly rather than by the use of pseudo-loads. Key issues in the formulation, such as non-linear coupling and growth of harmonic modes, are carefully and systematically explained. This coupled harmonics approach allows an easy implementation of the arc-length method. As a result, post-buckling load–deflection paths can be traced efficiently and accurately. The formulation also employs a non-linear shell theory more complete than existing classical theories. The results from the present study are independently verified using ABAQUS, while those from other studies are found to be inaccurate in general.     

Reliability-based structural optimization under hazard loads

The present paper studies the reliability-based structural optimization of the civil engineering in the seismic zone. The objective is to minimize the sum of construction material cost and the expected failure loss under severe earthquake, which is obtained by the sum of the products of the failure probability and its failure losses for the important failure modes. The set of constraints includes the deterministic constraints, and the constraints based on structural reliability—the reliability index constraints of structural element failure for the serviceability state under minor earthquake and the failure probability of the structural system for the ultimate limit state under severe earthquake. By introducing the load roughness index, the structural system reliability computation under hazard load can be greatly simplified, which is approximately determined by its weakest failure mode. Finally, the numerical example of high rising shear RC frame is calculated.     

Orthotropic cylindrical shells under line load along a generator

A technique for elastic analysis of an orthotropic cylindrical shell subjected to a uniform line load along a generator is developed. An accurate form of governing differential equations is derived and a mathematically discrete element method is used for its solution. The shell is divided into a finite number of longitudinal strips and the derivatives with respect to the circumferential coordinate in the governing equation are replaced by their finite difference relationships. The solution of the resulting equations is written in closed form. A computer program to implement this technique is developed and the computed results are compared with published experimental and analytical results. An excellent agreement is obtained. Some new results for a shell with fixed end boundary conditions are also presented.     

Buckling and postbuckling of imperfect cylindrical shells under axial compression

A solution methodology is described for the complete analysis of a geometrically imperfect, thin, circular, cylindrical shell loaded by a uniform axial compression. The analysis includes pre-limit point behavior, the establishment of critical conditions (limit point) and post-limit point behavior. The solution scheme is then utilized to study the effects of various geometrical parameters (radius to thickness and length to radius ratios) on the response characteristics of an imperfect, unstiffened, thin, cylindrical shell. These effects are assessed for a virtually axisymmetric-type of geometric imperfection.     

Optimization of geometrically imperfect stiffened cylindrical shells under axial compression

A procedure is outlined for optimizing stiffened, thin, circular, cylindrical shells under uniform axial compression against general instability, in the presence of initial geometric imperfection. The procedure consists of two parts (a) optimization on the basis of a linear buckling analysis and perfect geometry, and (b) parametric studies on a reasonable region in the design space surrounding the optimum point (as obtained from part (a)) to assess the effect of initial geometric imperfections. This procedure is demonstrated through two design examples, for which it is concluded, that the presence of initial geometric imperfections does not alter the optimum weight and the corresponding design variables appreciably.     

周期载荷下电活性聚合物圆柱壳的动力响应

基于非线性动力学理论研究了不可压电活性聚合物圆柱壳在内表面周期载荷作用下的运动与破坏等动力响应问题.通过对所得描述圆柱壳内表面运动的非线性常微分方程的数值计算和动力学定性分析,发现存在临界载荷和临界电压;当周期载荷的平均载荷值小于临界载荷及外加电压小于临界电压时,圆柱壳的运动随时间的演化是拟周期性的非线性振动.反之,圆柱壳将被破坏.讨论了外加电场和载荷参数对临界值和圆柱壳运动特性的影响.     

Buckling of cylindrical shells under the action of nonuniform external pressure

Over the last few decades, storage tanks have become bigger and thinner. Because of this, the buckling capacity of these cylindrical shells may well be the determining factor of shell thickness. In this paper, the critical buckling load of isotropic and orthotropic cylinders subjected to different types of wind load distributions is investigated. The prebuckling displacements are obtained by using the membrane theory of shell analysis. The principle of minimum potential energy in conjunction with Ritz's approach is used to obtain the stability matrix. The size of the stability matrix in this analysis is (81 × 81). By solving the stability matrix as an eigenvalue problem, the critical pressures are obtained as eigenvalues and the deflection shapes as eigenvectors. In the present study cylindrical shells of various dimensions, which are fixed at the base and free at the top, are investigated. The buckling load curves for isotropic and orthotropic cylinders of various dimensions are given for practical use.     

Interval-based optimization of aircraft wings under landing loads

An interval-based automated optimization of aircraft wing structures subjected to landing loads is discussed in this paper. The interaction between landing gear and flexible airplane structure is considered as a coupled system. The uncertain system parameters are described as interval numbers. The computational aspects of the optimization procedure are illustrated with two examples – symmetric double-wedge airfoil, and supersonic airplane wing. Since, in most cases only the ranges of uncertain parameters are known with their probability distribution functions unknown, the present methodology is expected to be more realistic for the optimum design of aircraft structures under landing loads.     

Collapse of long, noncircular, cylindrical shells under pure bending

This paper describes an extension of a method developed in a previous paper to determine the moment carrying capacity of elastoplastic noncircular cylindrical shells with infinite length by the finite element method. As a result of the shape change in the cross section of a shell during deformation, the bending moment reaches a global maximum value and then decreases as the bending curvature further increases. The shell would consequently collapse at the maximum moment. However, a bifurcation buckling may occur before the maximum moment can be developed. This bifurcation buckling could induce collapse of the shell under a moment less than the maximum. Determination of the likelihood that the bifurcation buckling would generate shell collapse may be made from the initial post-buckling behavior. An initial post-buckling analysis based on the J₂ deformation theory of plasticity has been developed in this paper. The finite element method with one spatial variable is used to locate the bifurcation point as well as to analyze the initial post-buckling behavior. Numerical examples of cylindrical shells with various cross-sectional shapes are shown. In particular, for a shell of square cross section, the moment at the bifurcation is much lower than the maximum value; however, the initial post-buckling analysis reveals that the state of equilibrium is still stable. Deep post-buckling analysis is required to determine the moment carrying capacity of a shell with such cross section.     

圆柱壳在热载荷及微扰外压作用下的分岔

在考虑温度对圆柱壳材料性能影响的基础上,建立了圆柱壳在扰动外压作用下的几何非线性动力控制方程.并采用伽辽金原理及 Melnikov 法研究了圆柱壳在热载荷及微扰外压作用下的分岔,进一步讨论分析了温度、Batdorf 参数等因素对圆柱壳发生混沌运动区域的影响,得出了随温度、Batdorf 参数的增大,混沌运动区域将越来越大的结论.     

Optimal design of cylindrical shells

In this paper, two types of problems of the optimal design of cylindrical shells with arbitrary axisymmetrical boundary conditions and distributed load, under the condition of the volume being constant, are discussed. These problems involve the minimax deflection and minimal compliancy of a cylindrical shell. Expressions of the objective function can be obtained by a stepped reduction method. In minimizing the maximum deflection, the position of the maximum deflection from the previous iteration is used as the next one. This procedure converges (Avriel 1976). Several examples are provided to illustrate the method.     

Structural optimization under uncertain loads and nodal locations

This paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes. The second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements. We first review the discrete form of the uncertain loads problem, which has been previously solved using a weighted average of multiple load patterns. With minor modifications, we extend this solution to include loads described by continuous joint probability density functions. We then proceed to the main contribution of this paper: structural optimization under uncertainty in the nodal locations. This optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix. It is shown, however, that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads. By expressing these equivalent loads in terms of continuous random variables, we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper. The optimization algorithms are developed in the context of minimum compliance (maximum stiffness) design. Simple examples are presented. The results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design. For structures containing thin substructures under axial loads, it is shown that these uncertainties (a) are of first-order significance, influencing the linear elastic response quantities, and (b) can affect designs by avoiding unrealistically optimistic and potentially unstable structures. The additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem.     

Analysis of shallow spherical and paraboloidal cantilever shells under symmetrically distributed loads

A mathematical stress and deformation analysis of thin shallow spherical and paraboloidal cantilever shells is presented. This paper describes a method of solution of Reissner's differential equations for the generally distributed loads, and then shows the determination of the particular solutions corresponding to rotationally symmetrical distributed loads. Computer results generated in the form of nondimensional curves for cantilever shells supported continuously around the apex by a rigid ring provide stresses and displacements in the most general form, allowing the engineer to perform an automatic stress analysis and to obtain an optimized design in a minimum length of time.     

Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion

This paper investigated the general instability of cylindrical shells in which the stiffeners formed spirals along the length and at an arbitrary angle with the axis. Two loading conditions were considered: uniform axial and lateral compressions and torsion. The stress-strain relations of the stiffeners were developed by rotation of the strain tensor. The buckling determinate was obtained by introducing into the equilibrium equations the admissible displacement functions consistent with the end constraints, thereby enforcing equilibrium by satisfying the characteristic equations.

The buclking equations were programmed for a computer which rearched through a finite set of stress resultants for assigned values of spiral angle and modes and printed out the buckling load. The optimum structure weight of the stiffened shell was determined by iterating the design parameters at the required spiral angle so that the buckling load approached the applied load as a limit until the difference between these loads was within the design allowance.     

Instability of short stiffened composite cylindrical shells under bending with prebuckling displacements

This paper presents results for buckling of a stiffened cylindrical shell with cutouts and both isotropic and composite shells without cutouts acting under end bending moments. The STAGS-C program has been used in the analysis.     

Computer simulation of the buckling behavior of thin shells under quasi static loads

Summary This paper concerns the theory and implementation of a numerical procedure that is capable of solving the collapse process of a structure when it is loaded past its ultimate carrying strength. The procedure is a combination of two generally available methods: A path following method for the quasi static part of the solutions and a transient method for the dynamic part. The simulation of a compression test on a thin walled cylindical shell with a local imperfection is provided to demonstrate the effectiveness of the strategy employed.     

Free vibration analysis of circular cylindrical shells

Determination of the vibrational characteristics of circular cylindrical shells often requires significant computational effort. This paper presents the results of a comprehensive, computer based, numerical investigation of the free vibration of circular cylindrical shells. An analytical procedure which accurately predicts the natural frequencies and radial mode shapes (corresponding to axial wave number and circumferential wave number both equal to one) for a wide range of circular cylindrical shells is developed. The procedure is applicable to shells either with or without a top closure. Several numerical examples are presented which illustrate application of the procedure and verify its accuracy.