A survey of buckling of conical shells subjected to axial compression and external pressure

The paper reviews literature on buckling of conical shells subjected to three loading conditions: (i) axial compression only, (ii) external pressure only and (iii) combined loading. The review is from the theoretical as well as experimental points of view. This review covers known experiments on cones from (1958 – 2012). The literature review is split thematically into the following categories: theoretical prediction of axially compressed cones, theoretical prediction of externally pressurized cones, theoretical prediction of cones under combined loading, buckling experiments on axially compressed cones, buckling experiments on externally pressurized cones, buckling experiments on cones subjected to combined loading, buckling experiments on composite conical shells, equivalent cylinder approach, effect of initial geometric imperfection on the buckling behaviour of cones and effect of imperfect boundary conditions on the buckling behaviour of cones

Static and Free Vibration Analyses of Composite Shells Based on Different Shell Theories

Equations of motion with required boundary conditions for doubly curved deep and thick composite shells are shown using two formulations. The first is based upon the formulation that was presented initially by Rath and Das (1973, J. Sound and Vib.) and followed by Reddy (1984, J. Engng. Mech. ASCE). In this formulation, plate stiffness parameters are used for thick shells, which reduced the equations to those applicable for shallow shells. This formulation is widely used but its accuracy has not been completely tested. The second formulation is based upon that of Qatu (1995, Compos. Press. Vessl. Indust.; 1999, Int. J. Solids Struct.). In this formulation, the stiffness parameters are calculated by using exact integration of the stress resultant equations. In addition, Qatu considered the radius of twist in his formulation. In both formulations, first order polynomials for in-plane displacements in the z-direction are utilized allowing for the inclusion of shear deformation and rotary inertia effects (first order shear deformation theory or FSDT). Also, FSDTQ has been modified in this dissertation using the radii of each laminate instead of using the radii of mid-plane in the moment of inertias and stress resultants equations. Exact static and free vibration solutions for isotropic and symmetric and anti-symmetric cross-ply cylindrical shells for different length-to-thickness and length-to-radius ratios are obtained using the above theories. Finally, the equations of motion are put together with the equations of stress resultants to arrive at a system of seventeen first-order differential equations. These equations are solved numerically with the aid of General Differential Quadrature (GDQ) method for isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with different boundary conditions using the above mentioned theories. Results obtained using all three theories (FSDT, FSDTQ and modified FSDTQ) are compared with the results available in literature and those obtained using a three-dimensional (3D) analysis. The latter (3D) is used here mainly to test the accuracy of the shell theories presented here

Static and Free Vibration Analyses of Composite Shells Based on Different Shell Theories

Equations of motion with required boundary conditions for doubly curved deep and thick composite shells are shown using two formulations. The first is based upon the formulation that was presented initially by Rath and Das (1973, J. Sound and Vib.) and followed by Reddy (1984, J. Engng. Mech. ASCE). In this formulation, plate stiffness parameters are used for thick shells, which reduced the equations to those applicable for shallow shells. This formulation is widely used but its accuracy has not been completely tested. The second formulation is based upon that of Qatu (1995, Compos. Press. Vessl. Indust.; 1999, Int. J. Solids Struct.). In this formulation, the stiffness parameters are calculated by using exact integration of the stress resultant equations. In addition, Qatu considered the radius of twist in his formulation. In both formulations, first order polynomials for in-plane displacements in the z-direction are utilized allowing for the inclusion of shear deformation and rotary inertia effects (first order shear deformation theory or FSDT). Also, FSDTQ has been modified in this dissertation using the radii of each laminate instead of using the radii of mid-plane in the moment of inertias and stress resultants equations. Exact static and free vibration solutions for isotropic and symmetric and anti-symmetric cross-ply cylindrical shells for different length-to-thickness and length-to-radius ratios are obtained using the above theories. Finally, the equations of motion are put together with the equations of stress resultants to arrive at a system of seventeen first-order differential equations. These equations are solved numerically with the aid of General Differential Quadrature (GDQ) method for isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with different boundary conditions using the above mentioned theories. Results obtained using all three theories (FSDT, FSDTQ and modified FSDTQ) are compared with the results available in literature and those obtained using a three-dimensional (3D) analysis. The latter (3D) is used here mainly to test the accuracy of the shell theories presented here

An empirical formula for predicting the collapse strength of composite cylindrical-shell structures under external pressure loads

This paper derives an empirical formula for predicting the collapse strength of composite cylindrical-shell structures under external hydrostatic pressure loads as a function of geometric dimensions and layered angles, where the effects of initial manufacturing imperfections are implicitly taken into account. A series of experiments are undertaken on [±θ/90] FW filament-wound-type composite cylindrical-shell models subjected to collapse pressure loads. A total of 20 composite cylindrical-shell models are tested to derive the empirical formula, which is validated by comparison with experimental data, existing design formulas of ASME 2007 and NASA SP-8700, and solutions of the nonlinear finite element method. It is concluded that the proposed formula accurately predicts the collapse pressure loads of filament-wound composite cylinders and will thus aid the safety design of composite cylindrical shell-structures under external pressure loads

Buckling Analysis and Optimum Design of Multidirectionally Stiffened Composite Curved Panel

Continuous filament grid-stiffened structure is a stiffening concept that combines structural efficiency and damage tolerance. However, buckle resistant design optimization of such structures using a finite element method is expensive and time consuming due to the number of design parameters that can be varied. An analytical optimization procedure which is simple, efficient and supports the preliminary design of grid-stiffened structures for application to combined loading cases is needed. An analytical model for a general grid-stiffened curved panel is developed using an improved smeared theory with a first-order, shear-deformation theory to account for transverse shear flexibilities and local skin-stiffener interaction effects. The local skin-stiffener interaction effects are accounted for by computing the stiffness due to the stiffener and the skin in the skin-stiffener region using the neutral surface profile of the skin-stiffener semi-infinite plate model. The neutral surface profile for the skin-stiffener semi-infinite plate model is obtained analytically using a stress function approach, minimum potential energy principle, and statics conditions. Analysis methods for buckling of general parallelogram-shaped and general triangular-shaped curved panels are developed. These analyses are required in order to assess the local buckling of grid-stiffened curved skin segments. The buckling analysis makes use of circulation functions as Ritz functions which account for material anisotropy and different boundary conditions. The local buckling of stiffener segments between stiffener interaction points are also assessed. Using these analyses and a genetic algorithm as optimizer, an optimization tool is developed for minimum weight design of composite grid-stiffened panel subjected to combined in-plane loads with a global buckling design constraint. Design variables are the axial and transverse stiffener spacings, the stiffener height and thickness, and the stiffener pattern. Results are presented for buckling loads of composite grid-stiffened panels which are obtained using the improved smeared theory and are compared with detailed finite element analysis. Buckling loads for anisotropic skewed and triangular plates, and curved panels are presented and compared with results from finite element analysis. Finally, designs for grid-stiffened panels obtained using the design optimization process are presented

ANALYTICAL STRIP METHOD FOR THIN CYLINDRICAL SHELLS

The Analytical Strip Method (ASM) for the analysis of thin cylindrical shells is presented in this dissertation. The system of three governing differential equations for the cylindrical shell are reduced to a single eighth order partial differential equation (PDE) in terms of a potential function. The PDE is solved as a single series form of the potential function, from which the displacement and force quantities are determined. The solution is applicable to isotropic, generally orthotropic, and laminated shells. Cylinders may have simply supported edges, clamped edges, free edges, or edges supported by isotropic beams. The cylindrical shell can be stiffened with isotropic beams in the circumferential direction placed anywhere along the length of the cylinder. The solution method can handle any combination of point loads, uniform loads, hydrostatic loads, sinusoidal loads, patch loads, and line loads applied in the radial direction. The results of the ASM are compared to results from existing analytical solutions and numerical solutions for several examples; the results for each of the methods were in good agreement. The ASM overcomes limitations of existing analytical solutions and provides an alternative to approximate numerical and semi-numerical methods

Finite Element Analysis of Geodesically Stiffened Cylindrical Composite Shells Using a Layerwise Theory

Layerwise finite element analyses of geodesically stiffened cylindrical shells are presented. The layerwise laminate theory of Reddy (LWTR) is developed and adapted to circular cylindrical shells. The Ritz variational method is used to develop an analytical approach for studying the buckling of simply supported geodesically stiffened shells with discrete stiffeners. This method utilizes a Lagrange multiplier technique to attach the stiffeners to the shell. The development of the layerwise shells couples a one-dimensional finite element through the thickness with a Navier solution that satisfies the boundary conditions. The buckling results from the Ritz discrete analytical method are compared with smeared buckling results and with NASA Testbed finite element results. The development of layerwise shell and beam finite elements is presented and these elements are used to perform the displacement field, stress, and first-ply failure analyses. The layerwise shell elements are used to model the shell skin and the layerwise beam elements are used to model the stiffeners. This arrangement allows the beam stiffeners to be assembled directly into the global stiffness matrix. A series of analytical studies are made to compare the response of geodesically stiffened shells as a function of loading, shell geometry, shell radii, shell laminate thickness, stiffener height, and geometric nonlinearity. Comparisons of the structural response of geodesically stiffened shells, axial and ring stiffened shells, and unstiffened shells are provided. In addition, interlaminar stress results near the stiffener intersection are presented. First-ply failure analyses for geodesically stiffened shells utilizing the Tsai-Wu failure criterion are presented for a few selected cases

Les effets des déformations en cisaillement et de l'inertie de rotation sur le comportement dynamique des coques nonumiformes, anisotropes et contenant un liquide en écoulement

Les théories classiques des coques -- Les effets des déformations de cisaillement dans l'analyse des plaques et des coques -- Étude de l'interaction dans un système couplé structure-fluide -- Les méthodes de solution -- General equations of anisotropic plates and shells including transverse shear deformations, rotatory inertia and initial curvature effects -- Transverse shear deformation in free vibration analysis of anisotropic open circular cylindrical shells -- Shear deformation in dynamic analysis of anisotropic laminated open cylindrical shells filled with or subjected to a flowing fluid

Finite element modelling and analysis of composite flywheel disk including effects of filament-winding mosaic pattern

A filament-wound spinning composite disk is characterised by the mosaic-patterned configuration of the layers produced during a filament-winding process. In this structure, each helically wound layer consists of curved triangular-shaped units alternating in the radial and circumferential directions. The mosaic-patterned configuration is not normally considered in the general stress analysis procedures based on the conventional modelling of laminated composite structures, including those available in FEA packages. However, the filament-winding mosaic pattern of the composite layer could significantly affect the stress fields developed due to rotational loading. Therefore, a methodology for the FE modelling and analysis of the filament-wound disk taking into account this effect is developed and structural analyses are performed using ANSYS, with different types of filament-winding mosaic patterns incorporated. Also, the disk is modelled and analysed using a conventional method and the differences in predicted stress values from both techniques are demonstrated through distributions of various stresses. The modelling governed by the first order shear deformation theory is performed using the SHELL 281 element. Firstly, using a conventional approach, the filament-wound composite disk is modelled as a laminated circular plate composed of different numbers of plies in which interlacing of plies due to filament-winding is not considered. Alternatively, three designs composed of 4, 8 and 14 plies are chosen to model the mosaic-patterned structure and to demonstrate changes in the stress levels in different layers and the extent of the influence of ply interlacing. Each design is associated with three types of mosaic-patterned configurations, namely 4, 6 and 8 mosaic units around the circumference of the disk. The disk is rotated at a constant angular velocity with the boundary conditions to prevent in-plane rigid body motions in both the radial and circumferential directions and also out-of-plane rigid body motion in the axial direction. As observed, the stress levels in the thin filament-wound composite flywheel disk could be underestimated in case of a structural analysis using the conventional mechanics of laminated structures. The layers of the filament-wound composite flywheel disk are reinforced with radially varying fibre trajectories that result in continuous changes in fibre orientation angles which generate stiffness variations and composite laminates with such stiffness variations are called variable-stiffness laminates. Thus, varying fibre trajectories should be modelled accurately to incorporate the actual stiffness variations for FEA of variable-stiffness composite structures. Therefore, a modelling approach is developed that would take into account the continuously varying fibre orientation angles derived from the predefined changing fibre trajectories. FE modelling of variable-stiffness laminates is performed using the proposed method and corresponding results obtained from various analyses are reported. Based on the results obtained from the numerical analyses of the filament-wound flywheel disk, various design aspects are assessed in terms of the dimensions and energy storage capacity of the disk. Parametric and comparative analyses of various disks are performed using different performance-controlling factors

Smart passive adaptive control of laminated composite plates (through optimisation of fibre orientation)

In the classical laminate plate theory for composite materials, it is assumed that the laminate is thin compared to its lateral dimensions and straight lines normal to the middle surface remain straight and normal to the surface after deformation. As a result, the induced twist which is due to the transverse shear stresses and strains are neglected. Also, this induced twist was considered as an unwanted displacement and hence was ignored. However, in certain cases this induced twist would not be redundant and can be a useful displacement to control the behaviour of the composite structure passively. In order to use this induced twist, there is a need for a modified model to predict the behaviour of laminated composites. A composite normally consists of two materials; matrix and fibres. Fibres can be embedded in different orientations in composite lay-ups. In this research, laminated composite models subject to transfer shear effect are studied. A semi analytical model based on Newton-Kantorovich-Quadrature Method is proposed. The presented model can estimate the induced twist displacement accurately. Unlike other semi analytical model, the new model is able to solve out of plane loads as well as in plane loads. It is important to mention that the constitutive equations of the composite materials (and as a result the induced twist) are determined by the orientation of fibres in laminae. The orientation of composite fibres can be optimised for specific load cases, such as longitudinal and in-plane loading. However, the methodologies utilised in these studies cannot be used for general analysis such as out of plane loading problems. This research presents a model whereby the thickness of laminated composite plates is minimised (for a desirable twist angle) by optimising the fibre orientations for different load cases. In the proposed model, the effect of transverse shear is considered. Simulated annealing (SA), which is a type of stochastic optimisation method, is used to search for the optimal design. This optimisation algorithm is not based on the starting point and it can escape from the local optimum points. In accordance with the annealing process where temperature decreases gradually, this algorithm converges to the global minimum. In this research, the Tsai-Wu failure criterion for composite laminate is chosen which is operationally simple and readily amenable to computational procedures. In addition, this criterion shows the difference between tensile and compressive strengths, through its linear terms. The numerical results are obtained and compared to the experimental data to validate the methodology. It is shown that there is a good agreement between finite element and experimental results. Also, results of the proposed simulated annealing optimisation model are compared to the outcomes from previous research with specific loading where the validity of the model is investigated