Zbigniew Kolakowski

Interactive buckling of thin-walled beam-columns with open and closed cross-sections

Abstract The influence of interactive buckling on the postbuckling behaviour of thin-walled elastic beams with imperfections is studied. The investigation is concerned with thin-walled closed and open cross-section beam-columns under axial compression and a constant bending moment. The beams are assumed to be simply supported at the ends. The asymptotic expansion established by Byskov and Hutchinson is employed in the numerical calculations in the form of the transition matrix method. The papers aim is to achieve the improved study of the equilibrium path in the postbuckling behaviour of imperfect structures with regard to the second-order approximation. The calculations are carried out for several types of beams.

Interactive buckling and load carrying capacity of thin-walled beam–columns with intermediate stiffeners

Abstract The design of thin-walled beam–columns must take into account the overall instability and the instability of component plates in the form of local buckling. This investigation is concerned with interactive buckling of thin-walled beam–columns with central intermediate stiffeners under axial compression and a constant bending moment. The columns are assumed to be simply supported at their ends. The asymptotic expansion established by Byskov and Hutchinson (AIAA J. 15 (1977) 941) is employed in the numerical calculations performed by means of the transition matrix method and Godunov’s orthogonalisation. Instead of the finite strip method, the exact transition matrix method is used in this case. The most important advantage of this method is that it enables us to describe a complete range of behaviour of thin-walled structures from all global (flexural, flexural-torsional, lateral, distortional and their combinations) to local stability. In the presented method for lower bound estimation of the load carrying capacity of structures, it is postulated that the reduced local critical load should be determined taking into account the global pre-critical bending within the first order non-linear approximation to the theory of the interactive buckling of the structure. The paper’s aim is to expand the study of the equilibrium path in the post-buckling behaviour of imperfect structures with regard to the second order non-linear approximation. In the solution obtained, the transformation of buckling modes with an increase of the load up to the ultimate load, the effect of cross-sectional distortions and the shear lag phenomenon are included. The calculations are carried out for a few beam–columns. The results are compared to those obtained from the design code and to the data reported by other authors. The results discussed in the present study represent the most important results obtained by the authors in earlier investigations devoted to central intermediate stiffeners (Int. J. Solid Struct. 32 (1995) 1501; Eng. Trans. 43 (1995) 383; Int. J. Solid Struct. 37 (2000) 3323; Int. J. Solid Struct. 33 (1996) 315; Thin Wall. Struct. 39 (2001) 649; Arch. Mech. Eng. XLVIII (2001) 29).

Modal interactive buckling of thin-walled composite beam-columns regarding distortional deformations

Abstract The influence of modal interactive elastic buckling on the post-buckling behaviour of thin-walled composite beam-columns with imperfections is studied when the distortional deformations are taken into account. The investigation is concerned with open cross-section columns under an axial compression or/and a constant bending moment. The beam-columns are assumed to be simply supported at the ends. The asymptotic expansion established by Byskov and Hutchinson [Mode interaction in axially stiffened cylindrical shells, AIAAJ, 15(7) (1977) 941–948] is employed in the form of the numerical transition matrix method. The papers aim is to improve the study of the equilibrium path in the post-buckling behaviour of imperfect structures with regarding the second order non-linear approximation and the effect of cross-sectional distortions. The principal goal of numerical analysis is to investigate the influence of the wall orthotropy factor of columns upon the all buckling modes from global (flexural, flexural–torsional, lateral, distortional and their combinations) to local and upon the coupled post-buckling state. In the solution obtained the transformation of buckling modes with the increase of load up to the ultimate load and shear lag phenomenon is included.

Interactive buckling of thin-walled beam-columns with intermediate stiffeners or/and variable thickness

Abstract The design of thin-walled beam-columns must take into account the overall instability and the instability of component plates in the form of local buckling. This investigation is concerned with interactive buckling of thin-walled beam-columns with central intermediate stiffeners or/and variable thickness under axial compression and a constant bending moment. The columns are assumed to be simply supported at the ends. The asymptotic expansion established by Byskov and Hutchinson (1977: Byskov, E. and Hutchinson, J. W. (1977). Mode interaction in axially stiffened cylindrical shells. AIAA J., 15(7), 941–948) is employed in the numerical calculations performed using the transition matrix method. The present paper is a continuation of the papers by Kolakowski and Teter (1995a,b: Kolakowski, Z. and Teter, A. (1995a). Interactive buckling of thin-walled closed elastic column-beams with intermediate stiffeners. Int. J. Solids Structures, 32(11), 1501–1516; Kolakowski, Z. and Teter, A. (1995b). Influence of local post-buckling behaviour on bending of thin-walled elastic beams with central intermediate stiffeners. Engineering Transactions, 43(3), 383–396) and Teter and Kolakowski (1996: Teter, A. and Kolakowski, Z. (1996). Interactive buckling of thin-walled open elastic beam-columns with intermediate stiffeners. Int. J. Solids Structures, 33(3), 315–330) where the interactive buckling of thin-walled beam-columns with central intermediate stiffeners in the first order approximation was considered. The paper’s aim is to improve the study of the equilibrium path in the post-buckling behaviour of imperfect structures with regard to the second order non-linear approximation. In the solution obtained the transformation of buckling modes with an increase of the load up to the ultimate load, the effect of cross-sectional distortions and shear lag phenomenon is included. The calculations are carried out for a few beam-columns.

Natural frequencies of a thin-walled structures with central intermediate stiffeners or/and variable thickness

Abstract The present paper deals with a dynamic analysis of stability of thin-walled beam-columns with central intermediate stiffeners or/and variable thickness when the shear lag phenomenon and the distortional deformations are taken into account. This investigation is concerned with thin-walled structures under axial compression and a constant bending moment. The structures are assumed to be simply supported at the ends. In order to obtain the equations of motion of individual plates, the non-linear theory of orthotropic thin-walled plates has been modified in such a way that it additionally accounts for all components of inertial forces. The differential equations of motion have been obtained from Hamilton’s principle, taking into account Lagrange’s description, full Green’s strain tensor for thin-walled plates and Kirchhoff’s stress tensor. The disturbance theory has been applied in order to obtain an approximate analytical solution to the equations. The problem of linear dynamic stability has been solved with the transition matrix method, taking into account Godunov’s orthogonalization. The calculations are carried out for a few beam-columns.

A semi-analytical method for the analysis of the interactive buckling of thin-walled elastic structures in the second order approximation

Abstract The investigation is concerned with a semi-analytical method for the interactive buckling analysis of thin-walled closed and open cross-section structures in the second order approximation based on the linear analysis. An improved “lower bound” approach by Koiter and Pignataro [Koiter, W. T. and Pignataro, M. (1976). A general theory for the interaction between local and overall buckling of stiffened panels. WTHD Report No. 556 , Delft University, p. 49] enables us to determine the overall flexural stiffness of a beam-column after its local buckling. The theoretical basis of the present approach is discussed and some typical examples are considered.

Interactive buckling of thin-walled closed elastic beam-columns with intermediate stiffeners

Abstract The design of thin-walled beam-columns must take into account the overall instability and the instability of component plates in the form of local buckling. This investigation is concerned with interactive buckling of thin-walled closed cross-section beam-columns with central intermediate stiffeners under axial compression and constant bending moment. The beams are assumed to be simply supported at the ends. The asymptotic expansion established by Byskov and Hutchinson is employed in the numerical calculations performed using the transition matrix method. The papers aim is to contribute to the study of the equilibrium path in the post-buckling behaviour of imperfect structures using the first order approximation. The calculations are carried out for a few closed beam-columns.

Lower bound estimation of load-carrying capacity of thin-walled structures with intermediate stiffeners

The design of thin structures must take into account the overall instability and the instability of component plates in the form of local buckling. This investigation is concerned with the interactive buckling of thin-walled structures with central intermediate stiffeners under axial compression and/or a constant bending moment. The structures are assumed to be simply supported at the ends. The lower bound estimation of load-carrying capacity on the basis of the post-buckling behaviour of thin-walled structures with imperfections is studied when the distortional deformations are taken into account. The asymptotic expansion established by Byskov and Hutchinson (AIAA J. 15 (1977) 941) is employed in the numerical calculations performed using the transition matrix method. The present paper is a continuation of previous work by the authors (Int. J. Solids Struct. 32 (1995) 1501; 33 (1996) 315; 37 (2000) 3323), where the interactive buckling of thin-walled beam-columns with central intermediate stiffeners in the first- and the second-order approximation were considered. In the solution obtained, the transformation of buckling modes with an increase in the load up to the ultimate load, the effect of cross-sectional distortions and the shear lag phenomenon are included. The results obtained are compared with data reported by other authors.

On some aspects of the modified TSAI-WU criterion in thin-walled composite structures

Abstract For anisotropic materials, Tsai-Wu [J Comp Mater 5 (1971) 58] proposed the failure criterion that takes into account the difference in strength due to positive and negative stresses. Hashin [J Appl Mech 47 (1980) 329] pointed to some internal incoherencies of the function of material effort with respect to stress parameters. The minimum of the External Function does not coincide with the unstressed state. Therefore, the criterion presented by Tsai-Wu [J Comp Mater 5 (1971) 58] has limited possibilities of application for optimization purposes. A modification of the Tsai-Wu criterion, needed in the case of the multi-criterion optimal design of thin-walled composite structures, has been presented. A proposal of the evaluation of the load carrying capacity of multi-layered composites with respect to their failure mode has been put forward.

Coupled Static and Dynamic Buckling Modelling of Thin-Walled Structures in Elastic Range Review of Selected Problems

Abstract A review of papers that investigate the static and dynamic coupled buckling and post-buckling behaviour of thin-walled structures is carried out. The problem of static coupled buckling is sufficiently well-recognized. The analysis of dynamic interactive buckling is limited in practice to columns, single plates and shells. The applications of finite element method (FEM) or/and analytical-numerical method (ANM) to solve interaction buckling problems are on-going. In Poland, the team of scientists from the Department of Strength of Materials, Lodz University of Technology and co-workers developed the analytical-numerical method. This method allows to determine static buckling stresses, natural frequencies, coefficients of the equation describing the post-buckling equilibrium path and dynamic response of the plate structure subjected to compression load and/or bending moment. Using the dynamic buckling criteria, it is possible to determine the dynamic critical load. They presented a lot of interesting results for problems of the static and dynamic coupled buckling of thin-walled plate structures with complex shapes of cross-sections, including an interaction of component plates. The most important advantage of presented analytical-numerical method is that it enables to describe all buckling modes and the post-buckling behaviours of thin-walled columns made of different materials. Thin isotropic, orthotropic or laminate structures were considered.