Chapter 8 Columns 255 Chapter 9 Thin Plates 294
8.5 Energy method for the calculation of buckling loads in columns
The fact that the total potential energy of an elastic body possesses a stationary value in an equilibrium state may be used to investigate the neutral equilibrium of a buckled
Fig. 8.14Shortening of a column due to buckling.
column. In particular, the energy method is extremely useful when the deflected form of the buckled column is unknown and has to be ‘guessed’.
First, we shall consider the pin-ended column shown in its buckled position in Fig. 8.14. The internal or strain energy U of the column is assumed to be produced by bending action alone and is given by the well known expression
U = l
0
M2
2EI dz (8.44)
or alternatively, since EI d2v/dz2= −M U = EI
2 l
0
d2v dz2
2
dz (8.45)
The potential energy V of the buckling load PCR, referred to the straight position of the column as the datum, is then
V = −PCRδ
whereδis the axial movement of PCR caused by the bending of the column from its initially straight position. By reference to Fig. 7.15(b) and Eq. (7.41) we see that
δ= 1 2
l
0
dv dz
2
dz giving
V = −PCR 2
l
0
dv dz
2
dz (8.46)
The total potential energy of the column in the neutral equilibrium of its buckled state is therefore
U+V= l
0
M2
2EI dz−PCR 2
l
0
dv dz
2
dz (8.47)
or, using the alternative form of U from Eq. (8.45) U+V= EI
2 l
0
d2v dz2
2
dz− PCR 2
l
0
dv dz
2
dz (8.48)
8.5 Energy method for the calculation of buckling loads in columns 273 We have seen in Chapter 7 that exact solutions of plate bending problems are obtain- able by energy methods when the deflected shape of the plate is known. An identical situation exists in the determination of critical loads for column and thin plate buckling modes. For the pin-ended column under discussion a deflected form of
v=∞
n=1
Ansinnπz
l (8.49)
satisfies the boundary conditions of (v)z=0 =(v)z=l =0
d2v dz2
z=0
= d2v
dz2
z=l
=0
and is capable, within the limits for which it is valid and if suitable values for the constant coefficients Anare chosen, of representing any continuous curve. We are therefore in a position to find PCRexactly. Substituting Eq. (8.49) into Eq. (8.48) gives
U+V = EI 2
l
0
π l
4∞
n=1
n2Ansinnπz l
2 dz
−PCR 2
l
0
π l
2∞
n=1
nAncosnπz l
2
dz (8.50)
The product terms in both integrals of Eq. (8.50) disappear on integration, leaving only integrated values of the squared terms. Thus
U+V = π4EI 4l3
∞ n=1
n4A2n−π2PCR 4l
∞ n=1
n2A2n (8.51) Assigning a stationary value to the total potential energy of Eq. (8.51) with respect to each coefficient Anin turn, then taking Anas being typical, we have
∂(U+V )
∂An = π4EIn4An
2l3 − π2PCRn2An
2l =0
from which
PCR= π2EIn2
l2 as before.
We see that each term in Eq. (8.49) represents a particular deflected shape with a corresponding critical load. Hence the first term represents the deflection of the column shown in Fig. 8.14, with PCR=π2EI/l2. The second and third terms correspond to the shapes shown in Fig. 8.3, having critical loads of 4π2EI/l2and 9π2EI/l2 and so on.
Clearly the column must be constrained to buckle into these more complex forms. In other words the column is being forced into an unnatural shape, is consequently stiffer and offers greater resistance to buckling as we observe from the higher values of critical
Fig. 8.15Buckling load for a built-in column by the energy method.
load. Such buckling modes, as stated in Section 8.1, are unstable and are generally of academic interest only.
If the deflected shape of the column is known it is immaterial which of Eqs (8.47) or (8.48) is used for the total potential energy. However, when only an approximate solution is possible Eq. (8.47) is preferable since the integral involving bending moment depends upon the accuracy of the assumed form ofv, whereas the corresponding term in Eq. (8.48) depends upon the accuracy of d2v/dz2. Generally, for an assumed deflection curvevis obtained much more accurately than d2v/dz2.
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 thatvis given by
v=A1f1(z)+A2f2(z)+A3f3(z)
Substitution in Eq. (8.47) results in an expression for total potential energy in terms of the critical load and the coefficients A1, A2and A3as the unknowns. Assigning stationary values to the total potential energy with respect to A1, A2and A3in turn produces three simultaneous equations from which the ratios A1/A2, A1/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. 8.15. An approximate shape may be deduced from the deflected shape of a tip-loaded cantilever.
Thus
v= v0z2
2l3 (3l−z)
This expression satisfies the end-conditions of deflection, viz.v=0 at z=0 andv=v0 at z=l. 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/dz2, is zero at the free end. The bending moment at any section is M=PCR(v0−v) so that substitution for M andvin Eq. (8.47) gives
U+V = P2CRv20 2EI
l
0
1− 3z2
2l2 + z3 2l3
2
dz− PCR 2
l
0
3v0 2l3
3
z2(2l−z)2dz