Buckling of circular cylindrical shells by the Differential Quadrature Method

A study is conducted of the linear elastic buckling of circular cylindrical shells by the new Differential Quadrature Method (DQM). To date this numerical method in the area of buckling analysis has been applied only to rectangular plates. The Fluegge shell stability equations serve as the basis of the analysis. By assuming the form of the buckling modes in the circumferential direction the stability equations are transformed into ones dependent only on the axial coordinate of the shell. The resulting ordinary differential equations are then solved using the one-dimensional DQM approach. Results are first given for shells with simple or clamped boundary conditions, and these are compared with previosly published results. Finally, new results are presented for shells with clamped-free boundary conditions, which have relevance to the buckling analysis of liquid storage tanks.     

The vibration and buckling of freely supported non-homogeneous orthotropic conical shells subjected to different uniform pressures

The paper examines the vibration and stability of freely supported orthotropic truncated and complete conical shells with non-homogeneous material properties under uniform lateral and hydrostatic pressures. First, the basic relations have been obtained for orthotropic truncated conical shells, Young's moduli and density which vary continuously in the thickness direction. By applying the Galerkin method, the buckling pressures and lowest cyclic frequencies of truncated and complete conical shells are obtained. To verify the accuracy of the results, comparisons are made with results available in the open literature. This study also examines in detail the effects of the variations of conical shell characteristics, the effects of the non-homogeneity and the orthotropy on the critical pressures and lowest cyclic frequencies of truncated and complete conical shells.     

Stability of a toroidal fluid-containing shell

This paper presents a solution to the stability problem of a sectorial toroidal shell subject to fluid loading. The solution is based on the Budiansky shell stability theory specialized for toroidal coordinates, and is developed using the two-dimensional differential quadrature method. Numerical results are presented and these results are compared with results from the Donnell shell theory, and with ones from a finite element method solution. The study establishes the suitability of the differential quadrature method for problems of shell stability.     

Elastic buckling of,and first yielding in,thin torispherical shells subjected to internal pressure

With the aid of the non-linear shell buckling computer program BOSOR 4, the internal pressures at which elastic circumferential buckling (or wrinkling) take place in thin torispherical shells have been calculated. The maximum equivalent (or effective) stresses in the shells in the axisymmetric pre-buckled state were also obtained; from these, the pressures at which first yielding in the shells commences were determined for $\text{1 <}\text{σ}{}_{\mathrm{<mtext>yp</mtext>}}\text{E}\text{× 10}{}^3\text{< 4}$The calculations were performed for shells with $\text{diameter}\text{thickness}$ ratios of 500, 1000 and 2000; other geometric ratios, as detailed in the paper, were also varied. The computations were carried out for steel shells but the results have been presented in dimensionless form.Utilising the above results it is possible to determine whether a given torispherical end closure will buckle elastically or whether an elastic-plastic analysis of the shell is desirable. Factors which are conducive to elastic buckling are a high yield point, a low modulus of elasticity or a large value of the shell diameter-thickness ($\text{D}\text{t}$) ratio. For steel shells, elastic internal pressure buckling will occur (for some combinations of $\text{r}\text{D}$ and $\text{R}{}_{\mathrm{<mtext>S</mtext>}}\text{D}$) for $\text{D}\text{t}\text{= 2000}$ and $\text{σ}{}_{\mathrm{<mtext>yp</mtext>}}\text{E}\text{= 3 × 10}{}^{\mathrm{?3}}$. For $\text{D}\text{t}\text{= 1000}\text{and}\text{500}$, first yield always precedes elastic buckling for the parameters investigated. The failure mode for these cases is either elastic-plastic buckling or plastic collapse (an axisymmetric mode with large deformations).A comparison of the results of linear and non-linear elastic axisymmetric stress analyses of the shells shows that the linear theory sometimes underestimates the first yield pressure by considerable amounts. Limit pressures obtained from small-deflection shell theories can be too low in such cases.Also given in the paper are approximate simple expressions whereby the elastic internal buckling pressures of torispherical shells may be calculated. These expressions should be useful to designers.     

Stress and stability analysis of toroidal shells

A finite element formulation applicable to the general shell of revolution is presented for the stress and stability analysis of toroidal pressure vessels under hydrostatic pressure. Considering the follower force effect of the external pressure, linear bifurcation buckling loads and corresponding mode shapes have been obtained in both axially and equatorially symmetric as well as antisymmetric buckling modes. Calculated critical values are compared with results of other investigations.     

Stresses and stability for the cone-cylinder shells with toroidal transition

The stresses in the cone-cylinder shells with a toroidal segment as a transition, subjected to external hydrostatic pressure, have been calculated by both analytical and finite element methods, and compared with those in cone-cylinder shells without tratisition. The stability of the shells with transition have been also analyzed by the Rayleigh-Ritz's method. The numerical results obtained show that the smooth shell formed by inserting a toroidal segment between the cone and cylinder, has much lower stresses and slightly higher buckling-loads than the corresponding cone-cylinder shell without transition.     

BUCKLING OF COMPOSITE CYLINDRICAL SHELLS UNDER MECHANICAL AND THERMAL LOADS

In this article the buckling of specially orthotropic laminated composite cylindrical shells is investigated. The nonlinear equilibrium equations based on the Sander's assumption of simplified nonlinear strain-displacement relations and the linearized stability equations for a circular cylindrical thin shell are considered. The equations include the rotations and transverse shear force. The resulting equations are improved compared to the Donnell stability equations. The mechanical and thermal buckling loads of a thin composite circular cylindrical shell based on Donnell and improved stability equations are obtained.     

A computer program for the elastostatics of a toroidal shell using the differential quadrature method

This paper presents a general analysis of the elastostatic problem of a sectorial toroidal shell. The solution to the problem is based on the Mushtari–Vlasov–Donnell shell theory and is developed using the two-dimensional quadrature method. A computer program incorporating the theory is attached, and an example of its application is given for the case of a fluid loading. Numerical results are determined and these are validated by a comparison with results obtained from a finite element solution.     

Stability analysis of a toroidal pipe-reducer under uniform external pressure

The stability of a toroidal pipe-reducer system is determined here from the solution of non-linear governing equations of axisymmetric deformations of shells of revolution. Numerical solutions are obtained by a modified version of the computer program developed by Uddin for solving the governing equations of axisymmetric shells by the multisegment method of integration. The interpretation of instability of the toroidal reducers is based on Thompson's theorems I and II. Critical pressures for the toroidal reduers are calculated over useful ranges of the curvature ratio, the thickness ratio, and the diameter ratio. It has been found that the critical pressure of these reducers varies almost linearly with the diameter ratio and that the long toroidal reducers are prone to local instability near the larger end. But this critical zone occurs near either one of the two ends as the reducer becomes shorter. The results of stability and stress analysis of toroidal pipe-reducers are compared here with those of conical reducers obtained by Ali and parabolic reducers obtained by Rahman. Comparison shows that toroidal reducers develop uniform stresses of lower magnitude compared to the other two. Further, toroidal reducers are found to sustain higher critical pressure than parabolic reducers except at higher diameter ratio.     

A new approach to instability testing of shells

In this paper, a new experimental technique for the buckling test of shells under external pressure is presented. This technique is simple and determines buckling load very efficiently. It has already been used extensively in the investigation of buckling load of electroformed sphericaltip conical shells under uniform external pressure. In this method, instead of the conventional monitoring of load and linear displacement of points on a shell, load and change in internal volume of the shell are recorded. Internal volumetric change of a shell is found to be a better indicator of its instability compared to the change in displacements of points on its surface. Volumetric change, a function of all the displacements and rotations contributing to the total change in its configuration, is observed to be a highly magnified indicator of shell deformation encompassing the whole shell, primarily the buckling zones and imperfection locations.     

THERMOELASTIC BUCKLING OF LAMINATED COMPOSITE CONICAL SHELLS

Within the framework of three-dimensional (3D) elasticity, an asymptotic theory is presented for the thermoelastic buckling analysis of laminated composite conical shells subjected to a uniform temperature change. A dimensionless parameter of thermal load related to the temperature change is defined. The method of perturbation is applied in the present formulation where the critical thermal loads and the primary field variables are expanded as a series of even powers of a small perturbation parameter. Through a straightforward derivation, the asymptotic formulation leads to recursive sets of governing equations for various orders. The classical shell theory is derived as a first-order approximation to the 3D theory. The method of differential quadrature is used to solve for the asymptotic solutions at each order level. The solvability conditions and normalization conditions for higher-order problems are derived. By considering these conditions, we can obtain the higher-order modifications. The critical thermal loads of simply supported, cross-ply conical shells are studied to demonstrate the performance of the present asymptotic theory.     

Two types of optimization of a cylindrical shell subjected to lateral pressure

Two types of optimization of thin-walled cylindrical shells loaded by lateral pressure are analyzed in this paper, with arbitrary axisymmetric boundary conditions and the volume being constant. The first is to find the optimal thickness to minimize the maximum deflection of a cylindrical shell. Here expressions of the objective function are obtained by the stepped reduction method. The optimal designs are reduced to nonlinear programming problems with an equality constraint. In minimizing the maximum deflection, the position of the maximum deflection from a previous iteration is used as the next one. The second is to find the optimal thickness to maximize the buckling pressure of shell. A buckling criterion of a shell is derived on the basis of an energy principle. An optimization criterion is formulated as the maximum of the buckling pressure. Moreover, the space of allowable solutions is defined. This procedure converges quickly and numerical results show the effectiveness of the method. Several examples are provided to illustrate the methods.     

开孔和补强对圆柱壳轴压屈曲的影响分析

  [目的]  早期的圆柱壳轴压屈曲试验结果与理论偏差较大,导致差异的因素较多,其中初始几何缺陷是产生差异的主要因素。  [方法]  文章采用一种专用的轴压屈曲试验平台,可以利用激光位移传感器扫描壳体三维实际形貌。屈曲试验轴压采用液压装置,轴压数据通过称重传感器读取。  [结果]  试验结果表明:开孔降低了圆柱壳轴压屈曲临界载荷,在开孔处插入圆管补强件,提高了圆柱壳轴压屈曲临界载荷。基于壳体形貌实测数据建立了有限元模型,利用Abaqus对圆柱壳、开孔圆柱壳和补强圆柱壳进行非线性屈曲有限元分析。  [结论]  模拟得到的规律与试验规律一致,壳体轴向载荷均是先呈线性增大后减小,开孔圆柱壳轴压屈曲临界载荷最小,含补强件圆柱壳轴压屈曲临界载荷次之,未开过孔的圆柱壳轴压屈曲临界载荷最大。对不同壳体的临界载荷模拟值与试验值进行了比较,其中最小的相对误差只有13.8%。说明运用此方法可以预测壳体轴压屈曲临界载荷,对壳体屈曲设计具有参考价值。     

Effects of Thermal Loading on the Buckling and Vibration of Ring-Stiffened Functionally Graded Shell

A theoretical method is developed to investigate the effects of thermal load and ring stiffeners on buckling and vibration characteristics of the functionally graded cylindrical shells, based on the first-order shear deformation theory (FSDT) considering rotary inertia. Heat conduction equation across the shell thickness is used to determine the temperature distribution. Material properties are assumed to be graded across the shell wall thickness of according to a power-law, in terms of the volume fractions of the constituents. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. The effects of stiffener's number and size on natural frequency of functionally graded cylindrical shells are investigated. Moreover, the influences of material composition, thermal loading and shell geometry parameters on buckling and vibration are studied. The obtained results have been compared with the analytical results of other researchers, which showed good agreement. The new features of thermal vibration and buckling of ring-stiffened functionally graded cylindrical shells and some meaningful and interesting results obtained in this article are helpful for the application and the design of functionally graded structures under thermal and mechanical loads.     

Experiment and numerical simulation of hydro-forming toroidal shells with different initial structure

In this paper, the effect of the initial structure on formation of toroidal shells is investigated for avoiding wrinkling in the hydro-forming process for manufacturing large elbows. The experimental results show that no wrinkling occurs on a toroidal shell with an octagonal cross-section. However, there are some tiny wrinkles on a toroidal shell with a hexagonal cross-section. The deformation process was simulated using an explicit dynamic finite element code LS-DYNA. The simulation results are basically in agreement with the experimental results. The reason for wrinkling is discussed. Research results show that wrinkling can be prevented by selecting an appropriate initial structure.     

Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods

The nonlinear dynamic response of doubly curved shallow shells resting on Winkler–Pasternak elastic foundation has been studied for step and sinusoidal loadings. Dynamic analogues of Von Karman–Donnel type shell equations are used. Clamped immovable and simply supported immovable boundary conditions are considered. The governing nonlinear partial differential equations of the shell are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The accuracy of the proposed HDQ-FD coupled methodology is demonstrated by numerical examples. The shear parameter G of the Pasternak foundation and the stiffness parameter K of the Winkler foundation have been found to have a significant influence on the dynamic response of the shell. It is concluded from the present study that the HDQ-FD methodolgy is a simple, efficient, and accurate method for the nonlinear analysis of doubly curved shallow shells resting on two-parameter elastic foundation.     

A three-dimensional layerwise-differential quadrature free vibration analysis of laminated cylindrical shells

A mixed layerwise theory and differential quadrature (DQ) method (LW-DQ) for three-dimensional free vibration analysis of arbitrary laminated circular cylindrical shells is introduced. Using the layerwise theory in conjunction with the three-dimensional form of Hamilton's principle, the transversely discretized equations of motion and the related boundary conditions are obtained. Then, the DQ method is employed to discretize the resulting equations in the axial directions. The fast convergence behavior of the method is demonstrated and its accuracy is verified by comparing the results with those of other shell theories obtained using conventional methods and also with those of ANSYS software. In the case of arbitrary laminated shells with simply supported ends, the exact solution is developed for comparison purposes. It is shown that using few DQ grid points, converged accurate solutions are obtained. Less computational efforts of the proposed approach with respect to ANSYS software is shown.     

Stresses at the junctions of axisymmetric shells under axially varying load

This paper deals with the investigation of stresses and deformations in the neighborhood of junctions of axisymmetric shells, made of segments of different geometries and subjected to different edge restraints and axially varying internal pressure. The method of investigation involves solution of a set of six first order highly nonlinear differential equations seeking the state of deformation of the shell at which, for a given pressure, the potential energy in the deformed shell is a relative minimum. The basic concept of the multisegment integration, as developed by Kalnins and Lestingi (On nonlinear analysis of elastic shells of revolution, J. Appl Mech. 1967;34:59–64), has been utilized for obtaining the solutions of the governing equations. Solutions for various problems are obtained for increasing loading, starting from a low value. The soundness of the method and the correctness of the computer program used in the analysis are verified by comparing the results with that of the corresponding analytical results available in the literature. Results are presented in the form of curves for stresses and deformations for varying loads and varying shell parameters which can be used for locating the critical zones and determining suitable values of shell parameters. The results include both stresses and deformations in their nondimensional forms based on both the linear and nonlinear theories.     

The buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler–Pasternak foundations

In this study, the buckling analysis of the simply supported truncated conical shell made of functionally graded materials (FGMs) is presented. The FGM truncated conical shell subjected to an axial compressive load and resting on Winkler–Pasternak type elastic foundations. The material properties of functionally graded shells are assumed to vary continuously through the thickness. The modified Donnell type stability and compatibility equations are solved by Galerkin’s method and the critical axial load of FGM truncated conical shells with and without elastic foundations have been found analytically. The appropriate formulas for homogenous and FGM cylindrical shells with and without elastic foundations are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. Finally, parametric studies on the buckling of FGM truncated conical and cylindrical shells on elastic foundations are being investigated. These parameters include; power-law and exponential distributions of FGM, Winkler foundation modulus, Pasternak foundation modulus and aspect ratios of shells.     

Asymmetric thermal buckling of annular plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behavior of isotropic homogeneous annular plates resting on a partial Winkler-type elastic foundation under uniform temperature elevation is investigated. First-order shear deformation plate theory is used to obtain the governing equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained under the solution of a plane stress formulation, neglecting the rotations and lateral deflection. Applying the adjacent equilibrium criterion, the linearized stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region, and the second set, which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalized differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved to obtain the critical conditions of the plate and the associated circumferential mode number. Benchmark results are given in tabular and graphical presentations for combinations of simply supported and clamped types of boundary conditions. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and the boundary conditions.