## 65 Buckling of columns panels and shells

(a) Isotropic solids

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 in Figure 6.4: 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 cases shown in the figure. Each is characterized by a value of the constant, n, which is equal to the number of half-wavelengths of the buckled shape.

Buckling of columns, panels and shells h*-i-H n

 n2n2EI fcrit _ f OR fcrit _ n2n2E A (*/r)2

F = Force (N) M = Moment (Nm) E = Youngs modulus (N/m2) e = Length (m) A = Section area (m2) I = See Figure 6.1 (m4) r = Gyration rad. ( A)2(m) k = Foundation stiffness (N/m2) n = Half-wavelengths in buckled shape p' = Pressure (N/m2)

F = Force (N) M = Moment (Nm) E = Youngs modulus (N/m2) e = Length (m) A = Section area (m2) I = See Figure 6.1 (m4) r = Gyration rad. ( A)2(m) k = Foundation stiffness (N/m2) n = Half-wavelengths in buckled shape p' = Pressure (N/m2)

M2 4 EI

Figure 6.4 Buckling of columns, panels and shells

The addition of the bending moment, M, reduces the buckling load by the amount shown in the second box in Figure 6.4. 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.

(b) Metal foams

The moduli of open-cell metal foams scale as (p/ps)2, that of closed-cell foams has an additional linear term (Table 4.2). When seeking elastic-buckling resistance at low weight, the material index characterizing performance (see Appendix) is E1/2/p (beams) or E1/3/p (panels). As beam-columns, foams have the same index value as the material of which they are made; as panels, they have a higher one, meaning that the foam panel is potentially lighter for the same buckling resistance. Sandwich structures with foam cores (Chapter 10) are better still. Clamping metal foams requires special attention: see Section 6.7.

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