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Buckling of Columns, Panels and Shafts

If sufficiently slender, an elastic column, loaded in compression, fails by elastic buckling at a critical load, Fcrit. This load is determined by the end constraints, of which four extreme cases are illustrated on Fig. A4: an end may be constrained in a position and direction; it may be free to rotate but not translate (or 'sway'); it may sway without rotation; and it may both sway and rotate. Pairs of these constraints applied to the ends of column lead to the five cases shown. Each is characterised by a value of the constant n which is equal to the number of half-wavelengths of the buckled shape.

The addition of the bending moment M reduces the buckling load by the amount shown in the second box. A negative value of Fcrit means that a tensile force is necessary to prevent buckling.

An elastic foundation is one that exerts a lateral restoring pressure, p, proportional to the deflection (p = ky where k is the foundation stiffness per unit depth and y the local lateral deflection). Its effect is to increase Fcrit, by the amount shown in the third box.

A thin-walled elastic tube will buckle inwards under an external pressure p', given in the last box. Here I refers to the second moment of area of a section of the tube wall cut parallel to the tube axis.

Figure A2 Buckling of Columns