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As the terms beam and column imply, this chapter deals with members whose crosssectional dimensions are small in comparison with their lengths. Particularly, we are concerned with the stability of beams and columns whose axes in the undeformed state are substantially straight. Classically, instability is associated with a state in which the deformation of an idealized, perfectly straight member can become arbitrarily large. However, some of the criteria for stable design which we will develop will take into account the influences of imperfections such as the eccentricity of the axial load and the crookedness of the centroidal axis of the column. The magnitudes of these imperfections are generally not known, but they can be estimated from manufacturing tolerances. For axially loaded columns, the onset of instability is related to the moment of inertia of the column cross section about its minor principal axis. For beams, stability design requires, in addition to the moment of inertia, the consideration of the torsional stiffness.

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We will begin with the familiar Euler column-buckling problem. The column is idealized as shown in Fig. 30.1. The top and bottom ends are pinned; that is, the moments

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at the ends are zero. The bottom pin is fixed against translation; the top pin is free to move in the vertical direction only; and the force P acts along the x axis, which coincides with the centroidal axis in the undeformed state. It is important to keep in mind that the analysis which follows applies only to columns with cross sections and loads that are symmetrical about the xy plane in Fig. 30.1 and satisfy the usual assumptions of linear beam theory. It is particularly important in this connection to keep in mind that this analysis is valid only when the deformation is such that the square of the slope of the tangent at any point on the deflection curve is negligibly small compared to unity (fortunately, this is generally true in design applications). In such a case, the familiar differential equation for the bending of a beam is applicable. Thus, EI For the column in Fig. 30.1, M = Py We take E and I as constant, and let P = k2 EI Then we get, from Eqs. (30.1), (30.2), and (30.3), d2y + k2 y = 0 dx2 (30.4) (30.3) (30.2) d2y =M dx2 (30.1)

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FIGURE 30.1 Deflection of a simply supported column. (a) Ideal simply supported column; (b) column-deflection curve; (c) free-body diagram of deflected segment.

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The boundary conditions at x = 0 and x = L are y(0) = y(L) = 0 (30.5)

In order that Eqs. (30.4) and (30.5) should have a solution y(x) that need not be equal to zero for all values of x, k must take one of the values in Eq. (30.6): k(n) = n L n = 1, 2, 3, . . . (30.6)

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which means that the axial load P must take one of the values in Eq. (30.7): P(n) = n2 2EI L2 n = 1, 2, 3, . . . (30.7)

For each value of n, the corresponding nonzero solution for y is y(n) = B(n) sin n x L n = 1, 2, 3, . . . (30.8)

where B(n) is an arbitrary constant. In words, the preceding results state the following: Suppose that we have a perfectly straight prismatic column with constant properties over its entire length. If the column is subjected to a perfectly axial load, there is a set of load values, together with a set of sine-shaped deformation curves for the column axis, such that the applied moment due to the axial load and the resisting internal moment are in equilibrium everywhere along the column, no matter what the amplitude of the sine curve may be. From Eq. (30.7), the smallest load at which such deformation occurs, called the critical load, is Pcr = This is the familiar Euler formula. 2EI L2 (30.9)

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