Buckling of a Tapered Column In Example 4 of Section 3.9, we saw that when a constant vertical compressive force or load P was applied to a thin column of uniform cross section, the deflection y(x) satisfied the boundary-value problem

The assumption here is that the column is hinged at both ends. The column will buckle or deflect only when the compressive force is a critical load p_n.

(a) In this problem let us assume that the column is of length L, is hinged at both ends, has circular cross sections, and is tapered as shown in FIGURE 5.2.1(a). If the column, a truncated cone, has a linear taper y = ex as shown in cross section in Figure 5.2.1(b), the moment of inertia of a cross section with respect to an axis perpendicular to the xy-plane is  we can write  Substituting J(x) into the differential equation in (24), we see that the deflection in this case is determined from the BVP

Where  Use the results of Problem 33 to find the critical loads P_n for the tapered column. Use an appropriate identity to express the buckling modes y_n(x) as a single function.

(b) Use a CAS to plot the graph of the first buckling mode y₁(x) corresponding to the Euler load P_1 when b = 11 and a= 1.

Example 4

Solve xy" + y = 0.

Problem 33

(a) The differential equation x⁴y'' + λy = 0 has an irregular singular point at x = 0. Show that the substitution t = 1/x yields the differential equation

which now has a regular singular point at t = 0.

(b) Use the method of this section to find two series solutions of the second equation in part (a) about the singular point t = 0.

(c) Express each series solution of the original equation in terms of elementary functions.