Euler buckling of columns 153

Structural instability

6.1 Euler buckling of columns 153

P

I

- I

I F

Displaced posit ion

P

posit ion

Fig. 6.1 Definition of buckling load for a perfect column.

there can be no sudden bowing or buckling. We therefore require a precise definition of buckling load which may be used in our analysis of the perfect column.

If the perfect column of Fig. 6.1 is subjected to a compressive load P, only shortening of the column occurs no matter what the value of P. However, if the column is displaced a small amount by a lateral load F then, at values of P below the critical or buckling load,

PCR,

removal of F results in a return of the column to its undisturbed position, indicating a state of stable equilibrium. At the critical load the displacement does not disappear and, in fact, the column will remain in any displaced position as long as the displacement is small. Thus, the buckling load

P C R is associated with a state of neutral equilibrium. For P

>

PCR enforced lateral displacements increase and the column is unstable.

Consider the pin-ended column AB of Fig. 6.2. We assume that it is in the displaced state of neutral equilibrium associated with buckling so that the compressive load

P

has attained the critical value PCR. Simple bending theory (see Section 9.1) gives

or

Fig. 6.2 Determination of buckling load for a pin-ended column

154 Structural instability

so that the differential equation of bending of the column is d2v PcR

-+-v=o

dzz

EI The well-known solution of Eq. (6.2) is

v = Acospz+ Bsinpz

where p2 = PcR/EI and A and B are unknown constants. The boundary conditions for this particular case are v = 0 at z = 0 and 1. Thus A = 0 and

Bsinpl= 0 For a non-trivial solution @e. v

#

0) then

sinpl=O or pl=n.rr wheren= 112131...

giving

or

Note that Eq. (6.3) cannot be solved for v no matter how many of the available boundary conditions are inserted. This is to be expected since the neutral state of equilibrium means that v is indeterminate.

The smallest value of buckling load, in other words the smallest value of P which can maintain the column in a neutral equilibrium state, is obtained by substituting n = 1 in Eq. (6.4). Hence

Other values of PCR corresponding to n = 2 , 3 , .

. .

are

These higher values of buckling load cause more complex modes of buckling such as those shown in Fig. 6.3. The different shapes may be produced by applying external restraints to a very slender column at the points of contraflexure to prevent lateral movement. If no restraints are provided then these forms of buckling are unstable and have little practical meaning.

PCR_

-

1/2

I -

PCH = 4 r 2 E I / L 2 P C R = ~ T ~ E I / L ' Fig. 6.3 Buckling loads for different buckling modes of a pin-ended column.

The critical stress, uCR, corresponding to P C R ,

2 E

( W 2

DCR = -

6.1 Euler buckling of columns 155 is, from Eq. (6.5)

(6.6) 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

and

(6.7)

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 L l l Boundary conditions

Both pinned Both fixed

One fixed, the other free One fixed, the other pinned

1 .o

0.5 2.0 0.6998

v = 0 at z = 0 and I

v = 0 at z = 0 and z = I. dvldz = 0 at z = I v = 0 and dv/d-. = 0 at z = 0

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

156 Structural instability

900

c N

E

z

a

600-

v

b"

300 Yield stress

A

-

-

- - - - - - I

I

^~ P=Pc, (bifurcation point)

0

Lateral deflection at mid-height

Fig. 6.4 Behaviour of a perfect pin-ended column.

theoretically possible for the column to take one of three deflection paths. Thus, if the column remains undisturbed the deflection at mid-height would continue to be zero but unstable (Le. the trivial solution of Eq. (6.3), u = 0) or, if disturbed, the column would buckle in either of two lateral directions; the point at which this possible branching occurs is called a bifurcation point; further bifurcation points occur at the higher values of P c R ( 4 ~ 2 E I / 1 2 , 9.ir2EI/12,. . .).

We have shown that the critical stress, Eq. (6.8), depends only on the elastic modulus of the material of the column and the slenderness ratio l / r . For a given material the critical stress increases as the slenderness ratio decreases; i.e. as the column becomes shorter and thicker. A point is then reached when the critical stress is greater than the yield stress of the material so that Eq. (6.8) is no longer applicable. For mild steel this point occurs at a slenderness ratio of approximately 100, as shown in Fig. 6.5.

t

I I I *

100 200 300 ( l / d

0 '

Fig. 6.5 Critical stress-slenderness ratio for a column.