Page 184 - Aircraft Stuctures for Engineering Student

168 Structural instability

I- 1

Fig. 6.13 Buckling load for a built-in column by the energy method.

Suppose that the deflection curve of a particular column is unknown or extremely
complicated. We then assume a reasonable shape which satisfies, as far as possible,
the end conditions of the column and the pattern of the deflected shape (Rayleigh-
Ritz method). Generally, the assumed shape is in the form of a finite series involving
a series of unknown constants and assumed functions of z. Let us suppose that v is
given by

w= A1fi(z) +-42fi(z) +A3h(Z)
Substitution in Eq. (6.47) results in an expression for total potential energy in terms of
the critical load and the coefficients Al, A2 and A3 as the unknowns. Assigning
stationary values to the total potential energy with respect to Al, A2 and A3 in turn
produces three simultaneous equations from which the ratios All& Al/A3 and
the critical load are determined. Absolute values of the coefficients are unobtainable
since the deflections of the column in its buckled state of neutral equilibrium are
indeterminate.
As a simple illustration consider the column shown in its buckled state in Fig. 6.13.
An approximate shape may be deduced from the deflected shape of a tip-loaded
cantilever. Thus

This expression satisfies the end-conditions of deflection, viz. v = 0 at z = 0 and
w = vo at z = 1. In addition, it satisfies the conditions that the slope of the column
is zero at the built-in end and that the bending moment, i.e. d2v/dz, is zero at the
free end. The bending moment at any section is M = PCR(v0 - w) so that substitution
for M and v in Eq. (6.47) gives

Integrating and substituting the limits we have

Hence