A Parametric Study on Geometrically Nonlinear Analysis of Initially Imperfect Shallow Spherical Shells

Abstract

Geometrically nonlinear analysis of initially imperfect shallow spherical shells under uniformly distributed axisymmetrical load is investigated in this computational study. The thickness of the shell is considered to be uniform and the material is assumed to be isotropic. The numerical treatment of the nonlinear fundamental shallow spherical shell equations is carried out by the finite difference method and the Newton—Raphson method. The influence of the parameters (Poisson's ratio, parameters of thickness, depth and initial imperfection) on the critical load, the value and the location of the maximum displacement components and the maximum stress resultants is examined by various numerical experiments each of which is made for two distinct types of supports along the edge: clamped and simply supported. Despite the dominant role of the nonlinearity, some general statements are obtained and the influence of the parameters is highlighted.

Nonlinear Behavior of Beams Having Initially Small Imperfection Subjected to Sinusoidal Load

“…Due to the convergence problem known to be occurring in the vicinity of the point where the slope of the force-deflection curve is zero [17][18][19]  is taken to be zero, then the state vector will be equal to zero vector. If a small increment is given to * n  , then the zero vector for the correction vector becomes a reasonable initial estimate for the Newton-Raphson iterations.…”

“…Due to the convergence problem known to be occurring in the vicinity of the point where the slope of the force-deflection curve is zero [17][18][19], the rotation at the nth point, * n  , is chosen to be removed from the unknown (state) vector and used in the input data, instead of 0 * q . 0 * q is put into the state vector as an unknown, in place of * n  .…”

Snap-through Buckling of Shallow Spherical Shells under Ring Loads

“…The terms (11), (12), (13) and (14) are inserted into equation (10) and expressed explicitly as follows [30]:…”

“…Although the snap-through behaviour of the relevant c structural elements resemble each other generally, the details of the concerning phenomenon (which is different for different structural elements, dimensional characteristics, materials used) can be crucially important from the point of design and analysis of such systems. Therefore, it is not a surprise to observe publications on this subject receiving much attention for many years, including those in recent years [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].…”

Effect of Compressibility on Nonlinear Buckling of Simply Supported Polyurethane Spherical Shells Subjected to an Apical Load

“…Most of the rubber-like materials have been assumed to be incompressible and there have been a host of studies about rubber-like incompressible shells, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], some of which have dealt with the behavior of incompressible spherical shells under apical loads.…”