Thickness Variation Parameter Research Articles

Modeling and Testing for Slotted Disk Springs Considering Linearly Gradient Thickness and Friction

The slotted disk spring has been widely used given its nonlinear characteristics. However, the number of studies on them is limited, and the influences of nonuniform thickness have not been investigated theoretically and experimentally. Additionally, the hysteresis caused by friction has been neglected in most studies. By introducing the symmetric friction condition and the dimensionless thickness variation parameter, the present study establishes a mathematical model for slotted disk springs with linearly graded thickness. By treating the slotted segment as a number of cantilever beams with both a linearly gradient thickness and width, a new analytical formula is developed to characterize the load–deflection of slotted disk springs based on the moment equilibrium equation. The proposed model allows the direct quantification of the influences of the thickness variation parameter and the symmetric friction condition on the load–deflection characteristics. To validate the proposed model, slotted disk springs with different parameter configurations are designed and tested. Comparisons between measured and theoretical results illustrate that the proposed model can effectively describe the load–deflection characteristics of slotted disk springs with both uniform thickness and linearly gradient thickness. Due to the hypothetic linear variable width of slotted segments, the proposed model has a better performance in predicting slotted disk springs with rectangular slots.

Elastic–plastic bifurcation analysis of cylindrical shells with axisymmetric variable thickness

This study focuses on the elastic–plastic bifurcation analysis of axial compressed thin-walled cylindrical shells with variable thickness. The general asymptotic formula of the elastic–plastic critical buckling load, which is a function of the Coefficient of the Thickness Variation (CTV) and the buckling load of shells with constant thickness, is derived based on the hybrid perturbation-weighted residual method. The general formula can be reduced to the classical elastic buckling solution in Koiter (1994) when considering the elastic case with the first-order term. Furthermore, a finite element bifurcation analysis procedure is also presented by adopting the incremental J2 deformation theory of plasticity through a custom user-defined subroutine appended to the nonlinear finite element software ABAQUS. The robustness of this numerical scheme and the theoretical formula is confirmed by the decent agreement with the available experimental results in the literature. In the end, the influence of the thickness variation parameter and D/t on the elastic–plastic buckling load of shells with varying thickness is also discussed.

Nonlinear transient dynamic analysis of laminated composite parabolic panels of revolution with variable thickness resting on elastic foundation

In this article, nonlinear transient behavior of laminated composite Parabolic Panels of Revolution Structure(s) (PPRS) with variable thickness resting on elastic foundation is investigated using Generalized Differential Quadrature (GDQ) method. Winkler-Pasternak model is used to represent elastic foundation. Linear, arch, sine and cosine thickness functions are used to express variable thickness. In transient analyses, First Order Shear Deformation Theory (FSDT) is used to consider the transverse shear effects. Nonlinearity is taken into account using Green-Lagrange nonlinear strain-displacement relations considering deepness effect. Virtual work principle is used to derive the equations of motion. Partial derivatives in the equation of motion are expressed with GDQ method and time integration is carried out using Newmark average acceleration method. Several problems are solved and compared with finite element results in order to validate the proposed method. After validation, effects of thickness functions, thickness variation parameter, geometric characteristic parameter of PPRS, boundary conditions, elastic foundation parameters as well as composite lamination scheme on nonlinear transient dynamic behaviour of PPRS are investigated.

Buckling of Cylindrical Shells with Arbitrary Circumferential Thickness Variations Under External Pressure

AbstractThis paper presents an analytical study on the buckling of cylindrical shells with arbitrary circumferential thickness variations under external pressure. Firstly, based on the thin shell theory and separation of variables, corresponding ordinary differential equations of laterally pressured cylinders are obtained. Secondly, the general asymptotic formula of buckling load, which is in terms of thickness variation parameter up to arbitrary order, is derived by combining the perturbation method and Fourier series expansion. Thirdly, the effects of uniform and circumferential modal thickness variations on the buckling of cylindrical shells under external pressure are investigated, respectively, and the results agree well with those available in literature. Particularly, the buckling load reduction of cylinders with circumferential modal thickness variation obtained by the proposed method coincides with the numerical results presented by Gusic et al. This analytical method is applicable for evaluating the stability of laterally pressured cylindrical shells with arbitrary circumferential thickness variations, once the thickness variation is known.

Dynamic Buckling of Cylindrical Shells with Arbitrary Axisymmetric Thickness Variation under Time Dependent External Pressure

This paper presents an analytical study on the critical dynamic buckling load of cylindrical shells with arbitrary axisymmetric thickness variation under uniform external pressure which is a function of time. Based on the Donnell simplified principle, the equilibrium and compatibility equations of cylindrical shells with arbitrary axisymmetric wall thickness under dynamic external pressure were derived. By using the method of separation of variables, the equations were transformed into ordinary differential equations in nondimensional form. Combining Fourier series expansion and the regular perturbation method, as well as the Sachenkov–Baktieva method, analytical formulas of the critical buckling load of cylindrical shells with arbitrary axisymmetric thickness variation under dynamic external pressure that varies as a power function of time were obtained. Using these analytical formulas, the critical dynamic buckling load of cylindrical shells with linearly and parbolically varying thickness were computed. The influences of thickness variation parameter and loading speed of external pressure on the critical buckling load were also discussed. The method was also applied to cylindrical shells with a classical cosine form thickness variation, by introducing the reduction factor of critical dynamic buckling load. The buckling capacity of these cylindrical shells under dynamic external pressure was discussed considering the effects of loading speed and thickness variation parameter.

An analytical method for the buckling analysis of cylindrical shells with non-axisymmetric thickness variations under external pressure

This article presents an analytical method for the buckling analysis of laterally pressured cylindrical shells with non-axisymmetric thickness variations. The previous results for thickness variations under external pressure are reviewed firstly. Then, a general analytical method that combines the perturbation method and Fourier series expansion is developed to derive buckling load formulas, which is in terms of thickness variation parameter up to arbitrary order. A classical non-axisymmetric thickness variation is discussed in detail by the presented analytical method. When non-axisymmetric modal thickness variation becomes axisymmetric, the buckling loads degenerate to the known results. Furthermore, the influence of circumferential modal thickness variation with mode corresponding to twice the circumferential buckling mode on the buckling of laterally pressured cylindrical shells is analytically investigated and the results show a great agreement with previous numerical ones by Gusic et al. Thus we confirm the presented method. In addition to theoretical analysis, calculations and comparisons are also performed. The general analytical method presented in the article can be utilized to determine the buckling loads of shells with general thickness variations.

Axisymmetric vibrations of layered cylindrical shells of variable thickness using spline function approximation

Free axisymmetric vibrations of layered cylindrical shells of variable thickness are studied using spline function approximation techniques. Three different types of thickness variations are considered namely linear, exponential and sinusoidal. The equations of axisymmetric motion of layered cylindrical shells, on the longitudinal and transverse displacement components are obtained using Love?s first approximation theory. A system of coupled differential equations on displacement functions are obtained by assuming the displacements in a separable form. Then the displacements are approximated using Bickley-spline approximation. The vibrations of two-layered cylindrical shells, made up of several types of layered materials and different boundary conditions are considered. Parametric studies have been made on the variation of frequency parameter with respect to the relative layer thickness, length ratio and type of thickness variation parameter.

Buckling of the initial imperfect rectangular thin plate with variable thickness

This paper analyzes the stability of the rectangular thin plate with sinusoidal changes in the plate thickness combined with initial curvature based on the large deflection theory. The buckling load for simply supported plates is defined using the energy method. The influence of the thickness variation parameter and the initial curvature parameter on the critical loads is investigated.

Stability of cylindrical panel with variable thickness

A linear buckling analysis based on the small deflection theory is presented for the cylindrical panel with sinusoidal changes in the shell thickness. The buckling load for simply supported cylindrical panel around the periphery is defined by using the hybrid perturbation - Galerkin method. The influence of the thickness variation parameter on the critical loads is investigated.

Free vibration of layered truncated conical shell frusta of differently varying thickness by the method of collocation with cubic and quintic splines

Free vibrations of layered conical shell frusta of differently varying thickness are studied using the spline function approximation technique. The equations of motion for layered conical shells, in the longitudinal, circumferential and transverse displacement components, are derived using extension of Love’s first approximation theory. Assuming the displacement components in a separable form, a system of coupled equations on three displacement functions are obtained. Since no closed form solutions are generally possible, a numerical solution procedure is adopted in which the displacement functions are approximated by cubic and quintic splines. A generalized eigenvalue problem is obtained which is solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The vibrations of two-layered conical shells, made up of several types of layer materials and supported differently at the ends are considered. Linear, sinusoidal and exponential variations in thickness of layers are assumed. Parametric studies are made on the variation of frequency parameter with respect to the relative layer thickness, cone angle, length ratio, type of thickness variation and thickness variation parameter. The effect of neglecting the coupling between bending and stretching is also analysed.

Axial buckling of composite cylindrical shells with periodic thickness variation

This study addresses the effect of thickness variation on the stability of the composite cylindrical shells under axial compression. Various lamination configurations and three commonly used composite materials are considered. Numerical results show that certain types of thickness variation patterns can greatly reduce the classical buckling load. Results indicate that the most detrimental effect of the thickness variation occurs when the wave number of the thickness variation is twice that of the classical buckling mode. For this case, an asymptotic formula is established by use of a computer algebra, which relates the buckling load reduction rate to the thickness variation parameter.

Buckling of an axially compressed cylindrical shell of variable thickness

This paper focuses on the buckling of cylindrical shells with small thickness variations. Two important cases of thickness variation pattern are considered. Asymptotic formulas up to the second order of the thickness variation parameter ε are derived by the combination of the perturbation and weighted residual methods. The expressions obtained in this study reduce to Koiter's formulas, when only the first-order term of the thickness variation parameter is retained in the analysis. Results from the asymptotic formulas are compared with those obtained through the purely numerical techniques of the finite difference method and the shooting method.

Vibration analysis of laminated conical shells with variable thickness

Effects of thickness variation on natural frequencies of laminated conical shells have been studied by using a semi-analytical finite element method. Love's first approximation thin shell theory is used to solve the problem. Effects of various parameters, such as the number of layers of the shell, the semi-vertex angle, the slant length to small end radius ratio and the thickness variation parameter (maximum to minimum thickness ratio), are studied, particularly on the lowest natural frequency. Shells of linear symmetrically varying thickness, about the mid-length of the shell, have been considered for the analyis. During the computation, the mass of the shell was kept constant for a particular slant length to small end radius ratio in order to provide useful examples of the effect of the thickness distribution on the natural frequencies.

Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic annular plates of variable thickness

Free axisymmetric vibrations of a polar orthotropic annular plate of linearly varying thickness resting on an elastic foundation of Winkler type are studied on the basis of classical theory of plates. The fourth order linear differential equation with variable coefficients governing the motion is solved by using the quintic spline interpolation technique for three different combinations of boundary conditions. The effect of the elastic foundation together with the orthotropy on the natural frequencies of vibration is illustrated for different values of the radii ratio and the thickness variation parameter for the first three modes of vibration. Transverse displacements and moments are presented for a specified plate. The validity of the spline technique is demonstrated by presenting a comparison of present results with those available in the literature.

Optimization study of anticlastic curvature of cylindrically bent strips with variable thickness

This paper presents a nonlinear extension of a previous investigation. Tapering the edges on the concave side of a longitudinally bent strip according to a power-law is studied aiming to alleviate the anticlastic deformations of the strip. Conjugate gradient method is employed to calculate the specific dimensions of power-law taperings which will reduce the peak-to-trough deflection of the cross-section of a bent strip to a minimum. Optimum tapering width and thickness ratios and thickness variation parameter are presented in graphic and tabular form. The influence of Poisson's ratio on the tapering configuration is discussed. The results show that power-law tapering can reduce the peak-to-trough deflection to less than 1 per cent of the thickness of the strip.