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.     

Frequency response of shell-plate combinations

Natural frequencies of cylindrical shells with a circular plate attached at arbitrary locations are determined for various boundary conditions and L/D ratios. The semi-analytical finite element method is used for the analysis. A conical shell element with four degrees of freedom per node and two nodes per element is used. For clamped-clamped and simply-supported boundary conditions the plate is attached at the center of the shell. For a clamped-free boundary condition the plate is at the free end of the shell. The effects of plate thickness and L/D ratio of the shell on the frequencies of the shell-plate combination are investigated.     

Application of penalty functions to a curved isoparametric axisymmetric thick shell element

A curved axisymmetric shell element with three nodes is developed. Quadratic interpolation is used and as the transverse shearing strain is included only first derivatives are required in the calculation of the strains. The element is found to yield accurate solutions for thick circular plates but a penalty factor must be used when the ratio of plate radius to thickness is of the order of 100. With appropriate values of the penalty factor, though, thin plate behaviour is reproduced with reasonable accuracy. Further, it is shown that for all practical purposes the penalty factor need only be based on the plate thickness. This is a useful conclusion in relation to shell analysis where different penalty factor values would otherwise need to be evaluated on the basis of the radius to thickness ratio. Finally the element is shown to give good results for cylindrical and spherical shells.     

Comparison of shell theories in the analysis of cylindrical shells with discontinuity in the thickness

Cylindrical shells with discontinuity in the thickness and that are subjected to axisymmetric loading have been analysed. Two types of finite elements are used: the first is based on thin shell theory and the second on thick shell theory. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. The effect of these loads for various end conditions and various step-ratios in the thickness have been analysed. Numerical results are presented and compared for both the theories. It has been shown that the transverse normal stress acting along the thickness direction is not negligible compared to other stresses at places of discontinuity either in the thickness or in the loading. The weight of the shell is kept constant for various step-ratios.     

Elastic buckling of double walled cylindrical storage tanks—numerical and analytical estimates

This paper is concerned with estimating the elastic buckling pressures of large liquid natural gas (LNG) storage tanks which are used by the British Gas Corporation for seasonal demand peak shaving. They consist of two concentric ring stiffened cylindrical shells separated by substantial thermal insulation which maintains the LNG within the inner shell at − 165°C with minimal boil off. There is natural gas vapour above the LNG and throughout the tank interior which is normally at just above atmospheric pressure. The shell walls increase in thickness from the top to the bottom and are fabricated from very thin steel or aluminium alloy plates (diameter to thickness ratio ~4000 at the top) since they are usually in hoop tension, but under certain conditions this can become compressive making elastic buckling a possible mode of failure. The individual buckling pressures for the two shells can be estimated using standard procedures but in these LNG tanks the annular insulation transfers loads between the shells enhancing their individual strengths. A numerical method using the finite difference code BOSOR4 and a simple analytical method have been used to estimate these pressures.     

Free vibration analysis of cylindrical liquid storage tanks

The circular cylindrical tank, built on grade and usually constructed in steel, aluminum or prestressed concrete, is one of the most common forms of liquid storage vessels. This paper summarizes the results of a comprehensive, computer based, numerical investigation of lateral free vibration of cylindrical liquid storage tanks. An analytical procedure which accurately predicts the fundamental frequencies (corresponding to axial wave number and circumferential wave number both equal to one) of a wide range of cylindrical storage tanks is developed. The procedure is applicable to tanks both completely full and partially full with liquid. Several numerical examples are presented which illustrate application of the procedure and verify its accuracy.     

A cylindrical tank-foundation-halfspace interaction using an energy approach

The interaction problem associated with a cylindrical storage tank-(plate) foundation system resting in frictionless contact on an isotropic halfspace is examined by using an energy formulation. The thin-plate and the thin-shell theories are considered in describing the flexural behavior of the foundation and the tank wall, respectively. The tank is assumed to be rigidly connected to the foundation. A power series in terms of the radial coordinate is used to represent the deflected shape of the plate. A variational technique is utilized to obtain the deflections and flexural moments at any point on the plate. The effects of interaction between various components of the tank-(plate) foundation-halfspace system are investigated in detail. A parametric nondimensional study shows that factors such as the relative stiffness and location of the tank wall may result in a more economical design of the (plate) foundation.     

A high precision triangular laminated anisotropic cylindrical shell finite element

The stiffness matrix for a high precision triangular laminated anisotropic cylindrical shell finite element has been formulated and coded into a composite structural analysis program. The versatility of the element's formulation enables its use in the analysis of multilayered composite plate and cylindrical shell type structures taking into account actual lamination parameters. The example applications presented demonstrated that accurate predictions of stresses as well as displacements are obtained with modest number of elements.     

Modeling and simulation of asymmetrical three-roll bending process

Asymmetrical three-roll bending is one of the three-roll bending processes widely used in metal forming due to its simple configuration. Modeling and simulation of the process based on finite element analyses has great interests to industrial practices. This paper deals with the prediction of the position of the lateral roll during cylindrical roll bending. In the numerical model, the rolls are assumed to be rigid bodies and the plate is assumed to be made of elasto-plastic material, with a bilinear material model corresponding to the results from tensile tests. Shell elements are applied to the plate and automatic node-to-surface contacts are selected for the interfaces between the plate and the rolls. The nonlinear equations are resolved by fully integrated Belytschko–Tsay shell formulation under the well-known ANSYS/LS-DYNA environment with explicit time integration. The positions of the lateral roll predicted by the numerical simulations agree well with experiments.     

Stability of cylindrical shell panels subjected to follower forces based on a mixed finite element formulation

A mixed finite element formulation is developed from a weak variational priniciple. This formulation is applied to stability analysis of cylindrical shell structures subjected to follower loading. Bilinear trial functions are used for all field variables. The rectangular curved elements presented here satisfy the continuity requirements for the field variables at the element interface. Two examples of a cantilevered cylindrical shell panel under different kinds of loading are solved.     

Geometrically nonlinear analysis of cylindrical shells using surface-parallel quadratic elements

A moderately thick cylindrical shell isoparametric element that is capable of accurately modeling cylindrically curved geometry, while also incorporating appropriate through-thickness kinematic relations is developed. The analysis accounts for fully nonlinear kinematic relations so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted. The present nonlinear finite element solution methodology is based on the hypothesis of linear displacement distribution through thickness (LDT) and the total Lagrangian formulation. A curvilinear side 16-node element with eight nodes on each of the top and bottom surfaces of a cylindrical shell has been implemented to model the transverse shear/normal deformation behavior represented by the LDT. The BFGS iterative scheme is used to solve the resulting nonlinear equations. A thin-shallow clamped cylindrical panel is investigated to test the convergence of the present element, and also to compare the special case of the present solution based on the KNSA (von Karman strain approximation) with those computed using the available faceted elements, discrete Kirchhoff constraint theory (DKT) and classical shallow shell finite elements, spanning the entire computed equilibrium path.     

Optimal shapes of cylindrical concrete water tanks

The optimal wall shapes of circular cylindrical concrete water tanks are treated herein. The tanks have piecewise linearly tapered wall thicknesses. The internal radius and height of the tank are maintained constant while the thickness of the wall is varied along the axis so that the critical bending and hoop stresses attain values as close as possible to their allowable values. For the analysis a procedure which combines the Runge-Kutta method of solution of ordinary differential equations with a numerical minimization method is used. This procedure is then imbedded into a minimization routine to deal with the design problems. The results of the study show that the major saving in material is achieved by one slope itself. While two slopes improve the saving in material for some tanks, it does not seem reasonable to increase the number of slopes beyond two.     

Incremental analysis of axisymmetric shallow shells with varying strain-displacement equations

The large displacement analysis of thin, shallow, axisymmetric shells is presented by using three different formulations for the strain-displacement relations. In the most general formulation all second-order terms of strains, rotations and their derivatives are taken into account; in the intermediary formulation the second-order terms of rotations only are accounted for; and in the third formulation the shallowness assumption is introduced and the Marguerre strain-displacement relations are adopted. The effectiveness of these formulations is investigated in numerical examples of a clamped circular plate and a clamped shallow spherical shell. Finally the stability of a clamped shallow spherical shell is examined using the Marguerre formulation between strains and displacements.     

Minimum cost design of a welded orthogonally stiffened cylindrical shell

In this study the optimal design of a cylindrical orthogonally stiffened shell member of an offshore fixed platform truss, loaded by axial compression and external pressure, is investigated. Ring stiffeners of welded box section and stringers of halved rolled I-section are used. The design variables considered in the optimization are the shell thickness as well as the dimensions and numbers of stiffeners. The design constraints relate to the shell, panel ring and panel stringer buckling, as well as manufacturing limitations. The cost function includes the cost of material, forming of plate elements into cylindrical shape, welding and painting. In the optimization a number of relatively new mathematical optimization methods (leap-frog - LFOPC, Dynamic-Q, ETOPC, and particle swarm - PSO) are used, in order to ensure confidence that the finally computed optimum design is accurately determined, and indeed corresponds to a global minimum. The continuous optimization procedures are adapted to allow for discrete values of the design variables to be used in the final manufacturing of the truss member. A comparison of the computed optimum costs of the stiffened and un-stiffened assemblies, shows that significant cost savings can be achieved by orthogonal stiffening, since the latter allows for considerable reduction of the shell thickness, which results in large material and manufacturing cost savings.     

A geometrically nonlinear shell finite element for tire vibration analysis

An axisymmetric finite element is developed which includes such features as orthotropic material properties, doubly curved geometry, and both the first and second order nonlinear stiffness terms. This element can be used to predict the equilibrium state of an axisymmetric shell structure with geometrically nonlinear large displacements. Small amplitude vibration analysis can then be performed based on this equilibrium state. The nonlinear path is predicted by using the self-correcting incremental procedure and any point on the path can be checked by using the Newton-Raphson iterative scheme. The present formulation and solution procedure are evaluated by analyzing a series of examples with results compared with alternative known solutions. Examples include: free vibration of an isotropic cylindrical shell, a conical frustum, and an orthotropic cylindrical shell; buckling of a cylindrical shell; large deflection of a clamped disk, a spherical cap, and a steel belted radial tire. The final example is a free vibration analysis of the inflated tire and the natural frequencies obtained compared well with published experimental data.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

Sensitivity analysis for the dynamic response of viscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric response of viscoplastic shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. Spatial discretization is performed using finite elements, with discontinuous stress resultants across element interfaces. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) with an implicit constitutive update. The sensitivity coefficients are evaluated using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading, and a circular plate subjected to impulsive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of these quantities at selected times.     

A unified approach to nonlinear buckling optimization of composite structures

A unified approach to nonlinear buckling fiber angle optimization of laminated composite shell structures is presented. The method includes loss of stability due to bifurcation and limiting behaviour. The optimization formulation is formulated as a mathematical programming problem and solved using gradient-based techniques. Buckling of a well-known cylindrical shell benchmark problem is studied and the solutions found in literature are proved to be incorrect. The nonlinear buckling optimization formulation is benchmarked against the traditional linear buckling optimization formulation through several numerical optimization cases of a composite cylindrical shell panel which clearly illustrates the advantage and potential of the presented approach.     

A unified approach for the analysis of bridges,plates and axisymmetric shells using the linear mindlin strip element

In this paper a finite strip formulation which allows to treat bridges, axisymmetric shells or plate structures of constant transverse cross section in an easily and unified manner is presented. The formulation is based on Mindlin's shell plate theory. One dimensional finite elements are used to discretize the transverse section and Fourier expansions are used to define the longitudinal/circumferential behavior of the structure. The element used is the simple two noded strip element with just one single integrating point. This allows to obtain all the element matrices in an explicit and economical form. Numerical examples for a variety of straight and curve bridges, axisymmetric shells and plate structures which show the efficiency of the formulation and accuracy of the linear strip element are given.     

A thin cylindrical shell finite element based on a mixed formulation

A specialized functional for thin cylindrical shells derived from the Washizu-Hu variational principle using considerations of relaxed continuity requirements is presented. A mixed formulation for a cylindrical thin shell finite element is developed from this functional. The assumed fields for displacements and stress resultants are bilinear functions in the longitudinal and circunferential directions.The agreement between the present results and those obtained in previous formulations is good if the comparison is based on the precision related to the number of variables involved in the problem.