Buckling of a Tapered Column In Example 4 of Section 3.9, we saw that when a constant

Chapter 5, Problem 34

(choose chapter or problem)

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

\(E I \frac{d^{2} y}{d x^{2}}+Py=0\), y(0)=0, y(L)=0

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= cx 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 \(I=\frac{1}{4} \pi r^{4}\), where r =y and y=cx. Hence we can write \(I(x)=I_{0}(x / b)^{4}\), where \(I_{0}=I(b)=\frac{1}{4} \pi(c b)^{4}\). Substituting I(x) into the differential equation in (24), we see that the deflection in this case is determined from the BVP

\(x^{4} \frac{d^{2} y}{d x^{2}}+\lambda y=0\), y(a)=0, y(b)=0

where \(\lambda=P b^{4} / E I_{0}\). 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 \(y_{1}\)(x) corresponding to the Euler load \(P_{1}\) when b= 11 and a= 1.