ESDU Engineer
Issue 08
Elasto-plastic buckling of flat isotropic stiffened panels and struts in compression
Data Item No. 01001 is accompanied by ESDUpac A0101 which provides a Fortran computer program that calculates initial elasto-plastic compressive buckling loads for stiffened rectangular panels and struts made of isotropic materials.
The program treats both elastic and plastic material behaviour and therefore basically represents an extension of the range of applicability of Data Item No. 98016, “Elastic buckling of flat isotropic stiffened panels and struts in compression”, into the plastic range of the material stress-strain curve.
The method on which the program is based is the finite strip method (FSM), which is particularly well-suited to the analysis of prismatic structures that show periodic deformation patterns under loading. In addition, although the program is based on FEM-like discretization procedures, it possesses significant advantages in speed compared to a full-scale finite element approach.
The purpose of the program is to automate and extend the range of applicability of the methods employed in various Data Items in the Aerospace Structures Series on the subject of buckling of elastic structures both in terms of the range of cross-section geometries that can be analysed and in terms of the treatment of plasticity.
Concerning the treatment of plasticity, three stress-strain curve models are provided in the program and available to the user. The first two models are variants of the curve originally proposed by Ramberg and
Osgood and the third model is another representation that is often found in the literature and which enables the specification of a strictly linear relationship between stress and strain below a particular stress-strain level. Full details of all three models and the parameters required to define them are given in the Item.
In the interest of quick and effective use, and in order to ease the input preparation process, the program is equipped with features that allow the user to specify a panel geometry using predefined skin-stringer sections, though the user may still generate the strip model data file for a panel of completely arbitrary cross-section without utilisation of the predefined sections.
The use of the predefined skin-stringer sections requires the listing of the typical section parameters, such as widths, thicknesses et cetera. Detailed information on the predefined sections is brought together in Appendix A of the Data Item.
An added option for use with the predefined skin-stringer sections is the possibility of representing a wide panel having many stringers by a special single skin-stringer model with predefined boundary conditions.
The special single-stringer model consists of a piece of skin of width equal to the stringer pitch with the stringer located centrally. The predefined boundary conditions have been selected in such a way as to give complete freedom in the z-direction whilst not allowing lateral in-plane displacements and rotations in the buckling mode of the outer edges of the skin section. This option allows quick exploration of the variation of buckling loads for ranges of geometrical and physical parameter values that may occur.
A model for a single stringer with user-specified boundary conditions can obviously be analysed as well.
In addition, adhesive layer and rivet line elements are provided to enable the joint between stringer and skin to be modelled in a more realistic way, though the capability to analyse inter-rivet buckling is not a feature of the program. Full details of the joint models are given in Appendix B of Data Item No. 98016.
Because of the inherent freedom, on account of the finite strip formulation, for the buckling modes to develop across the panel, it has been possible to extend the scope of the computable modes to include the combined torsional-flexural mode. No particular fixed points of rotation of the section, or parts of the section, are presumed to exist, other than those enforced by the user through imposition of boundary conditions. The boundary conditions at the longitudinal, or unloaded, edges are prescribed by the user; they may be free, simply supported or clamped.
Thus, ESDUpac A0101 is able to compute the elastic buckling stresses and associated modes for local, flexural and torsional or torsional-flexural buckling. However, the method presented has, as yet, no capabilities to analyse the response of a panel loaded under axial compression into the post-buckling range.
Three options for obtaining buckling loads are provided.
1. Calculation of the buckling load for the buckle half-wavelength equal to the specified panel length.
2. Calculation of the complete P–L curve with L ranging from a small value up to the actual panel length.
3. A more precise search, starting from a user-specified value, for a minimum in the load-length characteristic of the panel.
The computational capabilities of the program are demonstrated fully in the Item. Direct comparisons of results produced by the program with those given by other Data Items in the Structures Series are given, as are selected examples that illustrate the scope of the program and highlight its capabilities. Typical features that are inherent to the finite strip formulation are also outlined to give the user a broader insight into what is possible and what to expect when using program A0101.
Section 2 of the Data Item may be considered as a Program User’s Manual. It consists of descriptions of the input and output files, interwoven with appropriate examples to explain or elucidate detailed aspects of the finite strip input/output models.
Adam Quilter is the Group Head responsible for Structures. He can be contacted at aqui@esdu.com