1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 309

regardless of the boundary conditions, y(x) = 0 is a solution of equation ( 48).

The smallest value for the load P which will cause buckling is call ed the

critical load or the Euler load.

x

L

0

y(x)

Vertical

(Unbuckled)

Column

Buckled

Column

Figure 6.14 Buckling of a Column

EXAMPLE Finding the Critical Load for a Column

Find the critical load (Euler load) for a column which is clamped at the
base and pinned at the top.

SOLUTION

Let A^2 = P /EI. Then the auxiliary equation associated with equation ( 48)

is r^4 + A^2 r^2 = 0. The roots of this equation are r = 0, 0, Ai, and -Ai. So the
general solution of ( 48) is

y(x) =A+ Bx+ CsinAx + DcosAx.

Differentiating twice, we find

y' (x) = B +AC cos AX - AD sin AX and y"(x) = -A^2 C sin AX - A^2 D cos AX.

Since the column is clamped at the base, two boundary conditions are
y(O) = 0 and y' (0) = 0. And since the column is pinned at the top, two
additional boundary conditions are y(L) = 0 and y"(L) = 0. In order to
satisfy the boundary conditions, the constants A, B, C, and D must be chosen
to simultaneously satisfy the following four equations.