A Koiter-Newton arclength method for buckling-sensitive structures

Thin-walled structures, when properly designed, possess a high strength-to-weight and stiffness-to-weight ratio, and therefore are used as the primary components in some weight critical structural applications, such as aerospace and marine engineering. These structures are prone to be limited in their load carrying capability by buckling, while staying in the linear elastic range of the material. Buckling of thinwalled structures is an inherently nonlinear phenomena. When the material stays within its linear elastic range, the source of the nonlinearity is purely geometric. Thus, the analysis of nonlinear response of structures, especially thin-walled structures which are buckling sensitive, is important for determining their load carrying capability. For this reason, structural geometric nonlinearities are increasingly taken into account in engineering design. Nowadays, with the expanding computational power of modern computers nonlinear finite element analysis using commercial software is becoming the standard technique used to obtain the nonlinear response of complex structures, however, the repeated analyses that are needed in the design phase are still computationally intensive, in terms of the computation time required to run large models, even for modern computers. For this reason, reduced order techniques that reduce the problem size are attractive whenever repetitive analyses are required, such as in design optimization. Research on reduced order modeling of the nonlinear response of structures has attracted much attention from researchers. Some analytic techniques constitute very powerful tools for reducing the number of degrees of freedom (DOFs) in a nonlinear system, such as the Rayleigh-Ritz techniques and perturbation techniques. These two reduced basis techniques can be implemented in both analytical and numerical contexts, and due to the modeling versatility of the finite element method (FEM), most researchers prefer to reconstruct them within the FEM context, referred to as reduction methods. There are two families of reduction methods which can be recognized. The first family consists the path-following reduction methods which are based on some analytic techniques to reduce the number of DOFs in the full model and are able to trace the entire nonlinear equilibrium path of structures automatically, while they may find difficulties in the presence of buckling. Koiter reduction methods belong to the second family, and they are very good at handling the buckling sensitive cases due to the use of Koiter’s classical initial postbuckling theory, but the Koiter perturbation approach also limits the validity of these methods to a small range around the bifurcation point. The focus of the research reported in this thesis therefore is to find ways to synthesize the advantages of current reduction methods and obtain a new reduced basis path-following approach. In this thesis, a new approach called the Koiter-Newton (K-N) is presented for the numerical solution of a class of elastic nonlinear structural analysis problems. The method combines ideas from Koiter’s initial post-buckling analysis and Newton arclength methods to obtain an algorithm that is accurate over the entire equilibrium path of structures and efficient in the presence of buckling and/or imperfection sensitivity. The proposed approach is performed in a step by step manner to trace the entire equilibrium path, as is commonly used in the classical Newton arc-length method. In every expansion step, the approach works by combining a prediction step using a nonlinear reduced order model (ROM) based on Koiter’s initial postbuckling expansion with a Newton arc-length correction procedure. This nonlinear prediction provided by the reduced order model is much better compared to linear predictors used by the classical Newton-Raphson method, thus allowing the algorithm to use fairly large step sizes. The basic premise behind the proposed approach is the use of Koiter’s asymptotic expansion from the beginning rather than using it only at the bifurcation point in contrast to the traditional Koiter approaches. In each asymptotic expansion, the force space is reduced by the span of a set of perturbation loads that are chosen to excite the possible buckling branches. According to the stability of the equilibrium point, at which the asymptotic expansion is applied, different ways for selecting the perturbation loads are proposed. The proposed selection rules guarantee that the expansion step of the proposed approach can be applied at any point along the equilibrium path. The proposed technique requires derivatives of the element load vectors with respect to the degrees of freedom up to the third order. This is two orders more than what is traditionally needed for Newton’s method. To facilitate differentiation, nonlinear elements based on the element independent co-rotational frame are applied in the Koiter-Newton analysis. Automatic differentiation is used to find the derivatives of the co-rotational frame with respect to element degrees of freedom. In this way, full nonlinear kinematics are taken into account when constructing the reduced order model. In some cases, the nonlinear in-plane rotations of structures can be neglected, although the rotations of the normals to the mid-surface are finite. In such cases, Von Karman kinematics, which ignore some nonlinear items in the Green’s staindisplacement relations, possess an acceptable accuracy compared with the full nonlinear kinematics. Hence, the Koiter-Newton approach is also implemented based on Von Karman kinematics to achieve a better computational efficiency. Various numerical examples of beam and shell models are presented and used to evaluate the performance of the method. The Koiter-Newton analyses using the corotational kinematics and the Von K´arm´an kinematics are accurate and more computational efficient, compared with the results obtained using ABAQUS which adopts a full nonlinear analysis. The improved efficiency demonstrated by the Koiter-Newton technique will open the door to the direct use of detailed nonlinear finite element models in the design optimization of next generation flight and launch vehicles.Aerospace Structures and Computational MechanicsAerospace Engineerin