Analysis of ultimate limit state of spherical shell

Abstract

The article presents ultimate limit state analysis and limit load problem of a symmetrically loaded flat spherical shell. Physical parameters (modulus of elasticity, Poisson's ratio), shape, dimensions of the construction, load and its adding position and orientation are known. The mathematical model of the problem is formulated by technically computing the shells theory. The bending moments and axial forces are described by the second and the first degree polynomials. The element's differential statics equations, describing the balance between the internal and external forces, are replaced with algebraic equilibrium equations presented by the Bubnov-Galerkin method. The mathematical model and the calculation algorithm of the internal forces and displacements in the shell analysis problem are developed and formulated using statics and geometry equations. The construction is divided into countable elements, which are composed into a computational network. It is necessary to take into account not only the geometric shape of the structure, but also the distribution of load when the computational network of spherical shell is composed. The spherical shells are considered in the cylindrical (ρ,φ,z) co-ordinate system. The begining of the coordinate system is the construction center. The internal forces and the displacements are independent of j coordinates, when the load is symmetrical, so it is enough to investigate only one radial of the shell. The circular shell elements are connected by boundary nodes in the main nodes of the discrete model. The second-order circular element with three nodal (calculation) points in the one radial is used for discretization (Fig. 1). The mathematical model of elastic-plastic problem is a nonlinear mathematical programming problem.

Elastic internal forces S _e and displacements u _e are calculated by mathematical model (10)–(11). The values of internal forces and displacement of the main nodes are shown in Fig. 5. The values of nodal displacements are given up to the factor pR ₀ / E, while the values of the internal forces are given up to the factor pR ₀. The problem of limit load parameter p is calculated by mathematical model (15)–(16). The strength conditions are tested at all elements nodes. The value of limit load is p=2, 568 N ₀/R ₀. The Internal forces diagrams are shown Fig. 7. They are a corresponded plastic decomposition of flat spherical shell.

Santrauka

Darbe pateikiama simetriškai apkrauto lėkšto sferinio kevalo saugos ribinio būvio analizė. Konstrukcijos fizikiniai parametrai, forma, matmenys, apkrova ir jos pridėjimo vieta ir kryptis yra žinomi. Nagrinėjamas tamprusis ir idealiai tamprus plastinis kevalas, pasitelkiant ekstreminį energetinį principą randami tikrieji įtempių ir deformacijų būviai. Tampraus būvio analizės uždavinio matematinis modelis formuluojamas taikant virtualių jėgų principą, o ribinės apkrovos nustatymo uždavinio matematinis modelis formuluojamas kaip netiesinis matematinio programavimo uždavinys.

Reikšminiai žodžiai: lėkštasis sferinis kevalas, pusiausvirieji baigtiniai elementai, matematinis modelis, plastiškumo kriterijus, saugos ribinis būvis

Keyword : spherical shell, equilibrium finite element, a mathematical model of plasticity criteria, safety cut-off state