A10 - Introduction to Column Buckling1
Introduction of Column Buckling Equation Section 10
Structures subjected to compressive (and other types of loads) may become unstable and buckle. In idealized situations, buckling is the sudden onset of very large displacements at some critical load (generally transverse to the member) and sometimes with a corresponding decrease in load-carrying capacity. In other situations, buckling may occur more gradually; but as the load approaches the critical load displacements will increase at a rapid rate. Below are examples of buckling situations:
Consider a column fixed on one end and subjected to a uniaxial compressive load P. When P is small, the column shortens axially (is compressed). When the axial compressive force P reaches a critical value , the column suddenly experiences a lateral displacement, i.e., it buckles.
A thin, deep cantilever beam is subjected to a vertical end load P. As long as the load P is below a critical value , the beam section remains vertical (motion is downward only) and resists the bending action of P.
At the critical value , the beam will twist and bend sideward (out of the vertical plane).
The point at which the structure buckles is called an instability point. At or just below the critical value of the load, any small disturbance can cause the structure to change position as shown in the sketch of P vs. displacement.
A familiar soda can is shown below. When the applied load P is sufficiently small, the vertical wall remains cylindrical and is compressed uniformly in the vertical direction (fig. a).
If P becomes too large (reaches the critical value), the position becomes unstable. A small disturbance causes the vertical walls to bend in and out in a complex pattern as shown in fig. b (buckling or crumpling occurs). The top may even rotate relative to the bottom.
A somewhat different type of instability is shown below for a shallow curved arch or dome.
As the load P is increased, the top of the arch displaces downward in a somewhat linear fashion (fig. a).
However, at some critical value of P, the arch will suddenly snap through to the configuration shown in fig. b. This is called snap buckling. At this critical load, the arch (top) suddenly moves vertically from displacement A to B with NO increase in load P.
The investigation of structural instability and buckling is a difficult subject. We shall consider only the case of the cantilevered column discussed previously. Before considering this stability problem, it is necessary to derive the equations governing the bending of a beam subjected to longitudinal as well as transverse loads. Consider a free-body of a beam with a transverse load q(x) and a constant axial force P as shown below.
Summing forces vertically and taking moments about the center of the differential element yields:
(10.1)
Divide by and take the limit to obtain
(10.2)
Assume that the bending moment is responsible for the transverse deformation of the beam; i.e., we will neglect the effect of shear on the deformations (same as ENGR 214 and AERO 304). Then,
(10.3)
Substituting (10.3) into the moment equation (10.2) gives
(10.4)
Solving (10.4) for V and substituting in the shear equation (10.2) gives
(10.5)
Now consider the cantilevered column with only an axial compressive force P. Boundary conditions for this problem are given by:
(10.6)
The boundary conditions at x=L may be expressed in terms of by substituting the boundary conditions into the second of equations (10.2), and (10.3), into (10.6) to obtain:
(10.7)
For constant EI and P, the governing differential equation (10.5) becomes
(10.8)
We must now find the solution to the differential equation (10.8) subject to the boundary conditions at x=0 [eq. (10.6)] and x=L [eq. (10.7)]. We note that v=0 is a solution for any value of P. However, we are not interested in this trivial solution. The theory of differential equations states that we must have 4 independent constants in the general solution to the differential equation (there are 4 boundary conditions). A possible solution for is a combination of polynomial and trigonometric terms:
(10.9)
You can verify that this assumed solution satisfies the differential equation. Substituting (10.9) into the 4 boundary conditions [2 boundary conditons at x=0 in (10.6) and 2 at x=L in (10.7)] gives the following:
(10.10)
Note that all the right-hand sides are equal to 0; hence, a possible solution is that . In this case, is the solution for equilibrium of the cantilevered column. This would correspond to simple compression of the column with no sideways motion. However, we consider this once again a trivial solution. We need to find another solution!
Equations (10.10) are in fact an eigenvalue problem!
(10.11)
The solution of the eigenvalue problem requires that the determinant of the 4x4 coefficient matrix by equal to zero which will yield the solution for P satisfying this condition. Note that we will obtain an infinite number of solutions due to the repeating nature of the sin and cos trigonometric functions. An easier approach is as follows. Referring to equation (10.10), the fourth equation implies that is a possible solution (for ). With , the second equation implies that is a possible solution. The first equation implies that . Hence, the third equation becomes simply:
(10.12)
The last equation can be satisfied by setting , which is a trivial solution again, or by having a value of P such that
(10.13)
The smallest value of P satisfying this condition is
(10.14)
Substituting this value of P back into gives
(10.15)
Hence, we have found the critical value of P, and the shape that the beam bends into for this critical load. Note that the value of cannot be determined. This is the nature of an eigenvalue problem. Since the solution of an eigenvalue problem requires that we force the determinant of the coefficient matrix to be equal to zero, this is equivalent to making the equations linearly dependent. Linearly dependent equations can only be solved by assuming a solution for one (or more) of the unknowns (c's in this case); and the solution will always be in terms of the assumed c value. Note that when , the transverse deflection is zero. Transverse deflection occurs only when .
Hence, we have for the cantilevered column the critical value of P:
(10.16)
For other end conditions, we can follow the same procedure to obtain:
A10 - Introduction to Column Buckling1
A10 - Introduction to Column Buckling1
For axial loads that are not perfectly centered, we obtain an entirely different result. Consider the case when P is offset by an amount :
The problem may be worked as before, except that we treat the problem as having a perfectly centered load P plus a moment as shown above. We find that the third boundary condition in equations (10.10) is modified so that the right-hand side is equal to . Following the same procedure, we find that the transverse deflection is given by:
(10.17)
Plotting P vs. gives the plot on the right. For small values of P, the transverse deflection is very nearly zero. For example, when where is the value obtained for a perfectly centered load P on a cantilevered column. As P approaches the critical load, the deflection becomes very large. Because axial forces are rarely perfectly centered, one will always find some amount of transverse deflection occurring before P reaches the critical load.