4.6 Stability of Thinwall Structures
A critical factor in the design of thinwall structures is their stability. In much structural analysis, bending effects dominate and these are the critical stresses, however for thinwall structures, stability - resistance to buckling - is often crucial and all designs must be assessed for possible buckling failure which may be caused by compressive and / or shear loads.
The theory of panel instability under compression loading along opposite edges can be considered as an extension of that applying to a slender column. For a narrow 'strip' column, the strip width, 'b', can be used in the formula for the second moment of area for a rectangular cross section, I = b t³ / 12, for a strip thickness 't'. This value of 'I' is then used in Euler's formula. As the strip gets wider - and more like a panel - the effects of anticlastic bending (about an axis at 90^o to that of prime interest) have to be considered. The factor 1/(1-v²) then needs to be used.
In 1891 Bryan obtained a solution for the critical stress for in plane compressive loading of thin flat plates where the ratio: 'minimum width/thickness' is greater than 60, then the stress along b is given by:
stress = (pi² D a²/m² t)(( m² / a² )+( n² / b² )) where 'm' and 'n' are integers, 'D' is the flexural rigidity of a plate,
D = E t³ / (12 (1 - v² )) and 'pi' = 3.14159 and the plate is simply supported along all edges. The critical stress will always be a minimum for n = 1 but not necessarily for m = 1.
Graphs for different boundary conditions can be found in the literature.

Experimental work has shown that in practice the load is not carried uniformly accross the width of the plate, but is concentrated near the edges, giving rise to the sort of stress ditribution curve shown, right. The term 'effective width' is used to describe the fact that the load may be considered to be carried only by edge strips of the plate.

Local Instability
Rectangular box section, 'U' channels and other sections which comprise thin walls may be considered to be a collection of panels, which while they support each other along their (folded) edges, may still suffer buckling individually. As well as the stability of the overall section, the stability of each element of the section should be checked. Shear Instability
A common way of analysing a panel under shear buckling is to represent the panel as a framework of end loaded pin jointed bars having the panel within it or, braced by two diagonals bars. The mutual support of the panel and bars is represented adding a strip of skin 20 - 60 times its thickness to the bar cross section. It is assumed that the shear stress is uniform along the edge of the panel, although in reality it will be a maximum at the centre.

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