Rectangular Shallow Shells Research Articles

Stability Improvement of Flexible Shallow Shells Using Neutron Radiation.

Microelectromechanical systems (MEMS) are increasingly playing a significant role in the aviation industry and space exploration. Moreover, there is a need to study the neutron radiation effect on the MEMS structural members and the MEMS devices reliability in general. Experiments with MEMS structural members showed changes in their operation after exposure to neutron radiation. In this study, the neutron irradiation effect on the flexible MEMS resonators’ stability in the form of shallow rectangular shells is investigated. The theory of flexible rectangular shallow shells under the influence of both neutron irradiation and temperature field is developed. It consists of three components. First, the theory of flexible rectangular shallow shells under neutron radiation in temperature field was considered based on the Kirchhoff hypothesis and energetic Hamilton principle. Second, the theory of plasticity relaxation and cyclic loading were taken into account. Third, the Birger method of variable parameters was employed. The derived mathematical model was solved using both the finite difference method and the Bubnov–Galerkin method of higher approximations. It was established based on a few numeric examples that the irradiation direction of the MEMS structural members significantly affects the magnitude and shape of the plastic deformations’ distribution, as well as the forces magnitude in the shell middle surface, although qualitatively with the same deflection the diagrams of the main investigated functions were similar.

Напряженно-деформированное состояние прямоугольных в плане пологих оболочек переменной толщины при различных граничных условиях

Напряженно-деформированное состояние прямоугольных в плане пологих оболочек переменной толщины при различных граничных условиях

The equations of the loading process of shallow cylindrical shells under bilateral compression

The equations of the loading process in terms of rectangular shallow shells with bilateral compression when calculating the variational methods are derived. The problem is solved in view of geometrical non-linearity.

Solution describing the natural vibrations of rectangular shallow shells with varying thickness

Natural vibrations of shallow cylindrical shells with rectangular plan and varying thickness are studied using a spline-approximation method developed previously. Computation is carried out for different types of boundary conditions. The effect of the curvature of the midsurface on the natural frequencies is examined. The natural frequencies of shells with constant and varying thickness are compared

Moderately large vibrations of doubly curved shallow open shells composed of thick layers

This paper addresses nonlinear flexural vibrations of shallow shells composed of three thick layers with different shear flexibility, which are symmetrically arranged with respect to the middle surface. The considered shell structures of polygonal planform are hard hinged simply supported (i.e. all in-plane rotations and the bending moment vanish) with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first-order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. Numerical results of rectangular shallow shells in nonlinear steady-state vibration are presented for various ratios of shell rise to thickness, and non-dimensional load amplitude.

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.

Vertical wall surface forming of rectangular shell using multistage incremental forming with spherical and cylindrical rollers

A multistage incremental sheet metal bulging machine using spherical and cylindrical rollers has been developed for vertical wall surface forming of rectangular shallow shells. The multistage incremental forming comprises three operations: bulging with a spherical roller, right-angle forming and flattening with cylindrical rollers. The hand-operated bulging machine formed a vertical wall surface, which is highly precise and preferred by designers. A method of calculating for the approximate distribution of thickness strain and the maximum bulging height of the rectangular panel was proposed using the deformation limit diagram obtained from the incremental bulging test with a ball roller and a geometrical plane-strain deformation model with a constant strain gradient. The predictions for the rectangular shell were reasonably in good agreement with experimental values for the annealed aluminum sheet.

Nonlinear analysis of functionally graded plates and shallow shells

In this paper, an analytic solution is provided for the coupled large deflection of plates and shallow shells made of functionally graded materials (FGMs) under transverse mechanical loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of Fourier series. The effect of material properties, shell geometry and thermomechanical loading on the stress field are determined and discussed. The results reveal that thermomechanical coupling effects play a major role in dictating the response of the functionally graded shell.

Flexible and Incremental Sheet Metal Bulging using a Path-Controlled Spherical Roller. 2nd Report. Vertical Wall Surface Forming of Rectangular Shell Using Multistage Incremental Forming with Spherical and Cylindrical Rollers.

A multistage incremental sheet metal bulging machine using spherical and cylindrical rollers has been developed for vertical wall surface forming of rectangular shallow shells. Multistage incremental forming comprises three operations : bulging with a spherical roller, right-angle forming and flattening with cylindrical rollers. The hand-operated bulging machine formed a vertical wall surface which is highly precise and preferred by designers. A method of calculating for the approximate distribution of thickness strain and the maximum bulging height of the rectangular panel was proposed using the fracture limit obtained from the incremental bulging test with a ball roller and a geometrical plane-strain deformation model with a constant strain gradient. The predictions for the rectangular shell were in reasonably good agreement with experimental values for the annealed aluminum sheet.

A Ritz vibration analysis of doubly-curved rectangular shallow shells using a refined first-order theory

Integral expressions for strain and kinetic energies in vibration analysis of shear deformable doubly-curved shallow shells are presented. Although the formulation follows the first-order shear deformation theory, the consideration of Lamé parameters for the transverse shear strain through shell thickness, which have been hitherto neglected by other researchers, shows linear distribution functions instead of constant values if the Lamé parameters were dropped. Finite higher-order two-dimensional orthogonal polynomials ( pb-2 functions) previously used in thin shell analyses [3–6] have been generalized from three to five degrees to account for the additional rotation fields of a moderately thick shell. These global shape functions satisfy the kinematic boundary conditions at the outset. The excellent performance and versatility of the computational methodology of the pb-2 Ritz method are illustrated in representative numerical simulations. The resultant matrix size is far smaller than other discretization methods and yet accurate solutions can be obtained. The validity of the results is verified through direct comparisons. The discrepancy of various formulations with and without the Lamé parameters is presented and discussed.

Vibratory Characteristics of Cantilevered Rectangular Shallow Shells of Variable Thickness

In this paper, a comprehensive study on the modeling of vibratory response of variable thickness cantilevered shallow cylindrical shells of rectangular planform is carried out. Through the principle of minimum total energy, the equations of stretching and bending strain energies, kinetic energy, and the associated boundary conditions of the shells are derived. The present model is developed based on the Ritz formulation with the assumption of sets of mathematicall y complete admissible two-dimensional polynomials to approximate the in-plane and transverse displacement amplitude functions. A basic function is introduced in the approximate method to enforce automatic satisfaction of the kinematic boundary conditions. Sets of comprehensive reasonably accurate vibration frequencies of the shallow shells are presented for wide ranges of aspect ratio, thickness variation ratios, and shallowness ratio. These results, where possible, are verified by comparing with other established experimental and theoretical solutions. Few selected contour plots for the first known mode shapes of these cylindrical shells of variable thickness are included.

Numerical solution of multipoint boundary-value problems in improved shell theory

For some classes of stratified shells subject to transverse shear (shells of revolution with parameters varying along the generatrix, noncircular cylindrical shells with characteristics varying along the directrix, rectangular shallow shells), we propose an approach that reduces the solution of multipoint boundary-value problems to a number of two-point problems. As an example, we consider the stressed state of an open noncircular cylindrical shell supported in some section along the generatrix.