Buckling of an Elastic Column. An investigation of the

Chapter 11, Problem 26

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Buckling of an Elastic Column. An investigation of the buckling of a uniform elastic column of length L by an axial load P (Figure 11.2.1a) leads to the differential equationy(4) + y= 0, 0 < x < L.The parameter is equal to P/EI, where E is Youngs modulus and I is the moment of inertia of the cross section about an axis through the centroid perpendicular to the xy-plane. The boundary conditions at x = 0 and x = L depend on how the ends of the column are supported. Typical boundary conditions are = y= 0, clamped end,y = y= 0, simply supported (hinged) end.The bar shown in Figure 11.2.1a is simply supported at x = 0 and clamped at x = L. It is desired to determine the eigenvalues and eigenfunctions of Eq. (i) subject to suitable boundary conditions. In particular, the smallest eigenvalue 1 gives the load at which the column buckles, or can assume a curved equilibrium position, as shown in Figure 11.2.1b. The corresponding eigenfunction describes the configuration of the buckled column. Note that the differential equation (i) does not fall within the theory discussed in this section. It is possible to show, however, that in each of the cases given here all the eigenvalues are real and positive. 25 and 26 deal with column-buckling problems.In some buckling problems the eigenvalue parameter appears in the boundary conditionsas well as in the differential equation. One such case occurs when one end of the columnis clamped and the other end is free. In this case the differential equation y(4) + y= 0must be solved subject to the boundary conditionsy(0) = 0, y(0) = 0, y(L) = 0, y(L) + y(L) = 0Find the smallest eigenvalue and the corresponding eigenfunction.