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TRENDS IN CIVIL AND STRUCTURAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping, L.F. Costa Neves, R.C. Barros
Finite Strip Stability Solutions for General Boundary Conditions and the Extension of the Constrained Finite Strip Method
Z. Li and B.W. Schafer
Department of Civil Engineering, Johns Hopkins University, Baltimore, MD, United States of America
Z. Li, B.W. Schafer, "Finite Strip Stability Solutions for General Boundary Conditions and the Extension of the Constrained Finite Strip Method", in B.H.V. Topping, L.F. Costa Neves, R.C. Barros, (Editors), "Trends in Civil and Structural Engineering Computing", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 5, pp 103-130, 2009. doi:10.4203/csets.22.5
Keywords: finite strip method, finite element method, elastic buckling analysis, boundary conditions, constrained finite strip method.
The finite strip method (FSM) is a powerful and computationally efficient cousin of the more widely used finite element method (FEM). For thin-walled members, the finite strip method, in particular the semi-analytical FSM for simply supported boundary conditions as championed by Hancock  and others, has proved to be an organizing method for understanding all cross-section stability limit states. The laborious and subjective nature of identifying stability limit states in a shell finite element buckling analysis is also part of the reason for FSM’s popularity in thin-walled structures.
A limitation of widely available FSM implementations, such as CUFSM  is that they only apply to simply supported boundary conditions. Utilizing existing literature [3,4] the FSM elastic and geometric stiffness matrices for general end boundary conditions: pin-pin, fixed-fixed, fixed-pin, fixed-free, and fixed-guided are derived and provided in explicit form herein. Independent derivation is useful for fully understanding the method, and also leads to corrections in several terms from the published literature [3,4]. The key to the implementation is the introduction of specially selected trigonometric longitudinal series. The developed method is implemented in a custom version of CUFSM and validation studies are performed.
Validation studies on plates and members with general end boundary conditions are completed by comparing the FSM solutions to shell finite element eigenbuckling solutions. The length of the models is systematically varied such that local, distortional, and global buckling modes may be specifically examined. The results are excellent, indicating that the FSM solutions may readily provide solutions for general end boundary conditions. The one proviso is that the terms in the longitudinal series must be high enough to include all of the desired deformations. Specifically the number of buckled half-waves required to approximate local, distortional, and global buckling must be included in the longitudinal series.
Although the FSM results for general end boundary conditions are presented as a function of length, in the now classical FSM style (as in ), the interpretation of this curve is different. For simply-supported boundary conditions the solution may be fully represented as a function of buckling half-wavelength and all member lengths may be deduced from the solution for only one sinusoidal term. For general end boundary conditions the longitudinal terms participate with one another (illustrated through specific examples herein) and thus the solution is actually a function of physical length, not half-wavelength. In fact, conceptually, the modal identification problem in FSM with general boundary conditions essentially becomes the same as FEM.
To alleviate this modal identification problem and to provide an ability to decompose any stability solution into the relevant buckling classes: local, distortional, global – the second part of the paper provides derivations for the extension of the constrained FSM, or cFSM. The origins of cFSM [5,6,7] may be traced to General Beam Theory [8,9] and provide specific mechanical assumptions for the buckling classes. To date, these assumptions have only been implemented for simply supported boundary conditions. In the derivations presented herein, extension to general end boundary conditions is provided. Implementation of these derivations is currently under way.
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