Axial buckling of an orthotropic circular cylinder: Application to orthogrid concept

Abstract

A novel re-defining of the orthotropic material properties in terms of a so-called associated geometric mean isotropic (GMI) material is used to develop a thorough buckling analysis of an axially-loaded orthotropic circular cylinder. A membrane prebuckling condition is assumed and an expression for the buckling stress is derived in terms of cylinder geometry, orthotropic material properties, and the number of waves in the buckling deformation pattern in the axial and circumferential directions. By assuming the number of waves in each direction are real-valued variables, as opposed to integers, conditions which result in stationary values of the buckling stress are sought, and once found, examined for their character as regards representing minima, maxima, or saddle points. Three quite different buckling characteristics are predicted, the particulars depending on the shear modulus of the orthotropic material relative to that of the associated GMI material. It is shown that if the shear modulus of the orthotropic material is greater than the shear modulus of the associated GMI material, the cylinder buckles into a unique axisymmetric deformation pattern. If the shear modulus of the orthotropic material is less than the shear modulus of the associated GMI material, the cylinder buckles into a unique nonaxisymmetric deformation pattern. If the shear modulus of the orthotropic material is exactly equal to the shear modulus of the associated GMI material (this is the situation for an isotropic cylinder), the cylinder can buckle into either axisymmetric or nonaxisymmetric deformation patterns. Moreover it is shown that, in this case, there exists a number of deformation patterns, all at essentially the same stress level. Closed-form lower-bound expressions for the buckling stress are developed using the adopted notation, the value of the shear modulus relative to the shear modulus of the GMI material determining which expression is applicable. The results of this analysis are applied to a circular cylinder constructed of a lattice structure consisting of helical and circumferential ribs, a so-called orthogrid lattice cylinder, where it is assumed that the ribs of the lattice structure are dense enough to be able to represent the elastic properties of the lattice with an equivalent homogenized orthotropic material. An isogrid cylinder, where the helical rib angle is 30° relative to the axial direction, is a special case. The orthotropic cylinder analysis is reformulated in terms of the material properties of the ribs and the angle of the helical ribs. For this situation the isogrid case is the GMI material, and the rib angle determines whether the shear modulus of the equivalent orthotropic material is greater than or less than the GMI material. This translates into the character of the buckling deformations depending directly on the rib angle.