Semi-analytical tools for the analysis of laminated composite cylindrical and conical imperfect shells under various loading and boundary conditions

Abstract

The non-linear buckling of unstiffened laminated composite cones and cylinders will be investigated and new semi-analytical models capable to predict the static and the instability response of these shells under various loads and boundary conditions will be proposed. An introduction is given to the reader in order to present and discuss some of the main deterministic approaches currently used for the design of imperfection sensitive structures. From this introduction it will become clear the need for non-linear tools that can consider both geometric and load imperfections, which are recognized to be among the main factors affecting the load carrying capacity of the shells under discussion. The complete non-linear strain equations are derived using two Equivalent Single-Layer Theories: the Classical Laminated Plate Theory and the First-order Shear Deformation Theory. The non-linear terms will be identified corresponding to Donnell’s, Sanders’ and Timoshenko and Gere’ assumptions, but the discussion will focus on Donnell’s and Sanders’ equations. The resulting strain-displacement equations will then be applied to the stationary conditions of the total potential energy in order to obtain the non-linear static equations and the eigenvalue problem that can be used to predict the instability behavior, the latter using the neutral equilibrium criterion. The Ritz method is selected to solve the non-linear set of equations and a new set of appropriate approximation functions for the displacement field is proposed, in order to simulate axial compression, torsion, pressure, load asymmetry, any arbitrary surface or concentrated loads, and any load case combining these loads. Elastic constraints are used to produce a wide range of boundary conditions, covering the four types of boundary conditions commonly used in the literature. Two methods to solve the non-linear static equations are discussed: Newton-Raphson with line search and Riks; both presented in the full form, where the tangent stiffness matrix is calculated at every iteration, and in the modified form, where the tangent stiffness matrix is updated at the beginning of each load increment (or arc-length increment) and kept constant along the iterations. An analytical integration scheme is proposed for the linear stiffness matrices and a numerical integration scheme proposed for the non-linear stiffness matrices. The analytical integration schemes assume constant laminate properties over the whole conical/cylindrical surface. For conical shells a novel approximation is proposed in order to efficiently perform the analytical integration of the linear stiffness matrices. Detailed convergence analyses are presented and the proposed models are verified with finite element results, models available from the literature and test results from the literature. All the developed tools and algorithms are presented in detail and made available to the reader online. Keywords: Ritz method, Linear, Non-Linear, Static, Buckling, Composite, Conical, Cylindrical, Pressure, Torsion, Axial Compression, Donnell, Sanders, Geometric Imperfection, Load Imperfection