Geometrically nonlinear finite-element models for thin shells with geometric imperfections

Abstract

A finite-element method to analyze the stress-strain state and stability of thin shells with geometric imperfections is proposed. An arbitrary curvilinear finite element with vector approximation of the displacement function is used. To solve the systems of nonlinear algebraic equations by iteration methods, linearized stiffness matrices of finite elements and residual and load vectors are formed. The stress-strain state of a thin-walled shell with real geometric imperfections under surface pressure and axial compression is analyzed. The effect of geometric imperfections on the critical combination of loads is evaluated Introduction. Initial imperfections are the main factor that reduces the critical load. Therefore, development of methods for determining the critical loads of imperfect shells is still one of the major tasks of solid mechanics [1-3, 5, 6, 9, 11, 12]. Donnell was one of the first to analyze the sensitivity of the critical load to initial geometric imperfections [10]. Of great importance for the theory of stability of imperfect shells is Koiter's asymptotic method [8], which allows us to evaluate the sensitivity of structures to imperfections from the value of the first zero coefficient in the power dependence of the critical load on the amplitude of the bifurcation mode. It is assumed that the effect of imperfections, which are generally random and separated as bifurcation modes, is mainly determined by one component similar to the lowest instability mode. Koiter's method is mainly used to analyze cylindrical and spherical shells, which are characterized by poor agreement between experimental and theoretical results. There are also approaches [4,5,13] that incorporate the initial imperfections of shells into the geometric parameters of the design model and solve the nonlinear problem numerically.The finite-element method is widely used to analyze the stress-strain state and stability of shells [7]. The results of stability analysis of thin shells with real imperfections modeled by buckling modes are presented in [4,5,13]. In studying shells with real imperfections, a shell of canonical geometry is first generated and then imperfections are imposed on its surface. When imperfections are modeled by buckling modes, a finite-element model with smooth surface is first generated and then a linear buckling problem is solved. To evaluate the effect of the amplitudes of imperfections on the critical load, a new finite-element model of the shell is obtained by correcting its geometry. This is why setting up geometrically nonlinear models of imperfect shells with the methods mentioned above is labor-intensive and time-consuming. One more disadvantage of these methods is that only triangular plane finite elements can be used to model the shell surface.The present paper proposes a method to construct geometrically nonlinear finite-element models of thin shells with geometric imperfections based on an approach that was earlier used in analytical methods for solving such problems. The finite-element equations w...

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