Page 171 - Aircraft Stuctures for Engineering Student

6.1 Euler buckling of columns 155

The critical stress, uCR, corresponding to PCR, is, from Eq. (6.5)
2E
DCR = - (6.6)
(W2
where r is the radius of gyration of the cross-sectional area of the column. The term
l/r is known as the slenderness ratio of the column. For a column that is not doubly
symmetrical, r is the least radius of gyration of the cross-section since the column will
bend about an axis about which the flexural rigidity EI is least. Alternatively, if
buckling is prevented in all but one plane then EI is the flexural rigidity in that plane.
Equations (6.5) and (6.6) may be written in the form

(6.7)
and

where I, is the efective length of the column. This is the length of a pin-ended column
that would have the same critical load as that of a column of length 1, but with
different end conditions. The determination of critical load and stress is carried out
in an identical manner to that for the pin-ended column except that the boundary
conditions are different in each case. Table 6.1 gives the solution in terms of effective
length for columns having a variety of end conditions. In addition, the boundary
conditions referred to the coordinate axes of Fig. 6.2 are quoted. The last case in
Table 6.1 involves the solution of a transcendental equation; this is most readily
accomplished by a graphical method.

Table 6.1
Ends Lll Boundary conditions
Both pinned 1 .o v= 0 at z = 0 and I
Both fixed 0.5 v = 0 at z = 0 and z = I. dvldz = 0 at z = I
One fixed, the other free 2.0 v = 0 and dv/d-. = 0 at z = 0
One fixed, the other pinned 0.6998 dvldr = 0 at I’ = 0, v = 0 at z = 1 and z = 0
~~~~ ~ ~ ~ ~~ ~ ~~ ~ ~~ ~
Let us now examine the buckling of the perfect pin-ended column of Fig. 6.2 in
greater detail. We have shown, in Eq. (6.4), that the column will buckle at discrete
values of axial load and that associated with each value of buckling load there is a
particular buckling mode (Fig. 6.3). These discrete values of buckling load are
called eigenvalues, their associated functions (in this case Y = Bsinnm/l) are called
eigenfunctions and the problem itself is called an eigenvalue problem.
Further, suppose that the lateral load F in Fig. 6.1 is removed. Since the column is
perfectly straight, homogeneous and loaded exactly along its axis, it will suffer only
axial compression as P is increased. This situation, theoretically, would continue
until yielding of the material of the column occurred. However, as we have seen,
for values of P below PcR the column is in stable equilibrium whereas for P > PCR
the column is unstable. A plot of load against lateral deflection at mid-height
would therefore have the form shown in Fig. 6.4 where, at the point P = PCR, it is