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The critical-load formulas developed above provide satisfactory values of the allowable load for very slender columns for which buckling, as manifested by unaccept-

Figure 3-16

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ably large deformation, will occur within the elastic range of the material. For more massive columns, the deformation enters the plastic region (where strain increases more rapidly with stress) prior to the onset of buckling. To take into account this change in the stress-strain relationship, we modify the Euler formula. We define the tangent modulus E(t) as the slope of the tangent to the stress-strain curve at a given strain. Then the modified formulas for the critical load are obtained by substituting E(t) for E in Eq. (30.9) and Eq. (30.13) plus Eq. (30.14) or Eq. (30.16) plus Eq. (30.14). This will produce a more accurate prediction of the buckling load. However, this may not be the most desirable design approach. In general, a design which will produce plastic deformation under the operating load is undesirable. Hence, for a column which will undergo plastic deformation prior to buckling, the preferred design-limiting criterion is the onset of plastic deformation, not the buckling.

We will now develop a design criterion which will enable us to use the yield strength as the upper bound for acceptable design regardless of whether the stress at the onset of yielding precedes or follows buckling. Here we follow Ref. [30.1]. This approach has the advantage of providing a single bounding criterion that holds irrespective of the mode of failure. We begin by noting that, in general, real columns will have some imperfection, such as crookedness of the centroidal axis or eccentricity of the axial load. Figure 30.4 shows the difference between the behavior of an ideal, perfectly straight column subjected to an axial load, in which case we obtain a distinct critical point, and the behavior of a column with some imperfection. It is clear from Fig. 30.4 that the load-deflection curve for an imperfect column has no distinct critical point. Instead, it has two distinct regions. For small axial loads, the deflection increases slowly with load. When the load is approaching the critical value obtained for a perfect column, a small increment in load produces a large change in deflection. These two regions are joined by a knee. Thus the advent of buckling in a real column corresponds to the entry of the column into the second, above-the-knee, load-deflection region. A massive column will reach the stress at the yield point prior to buckling, so that the yield strength will be the limiting criterion for the maximum allowable load. A slender column will enter the above-the-

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DI-0 DI-1 DI-2 DI-3

knee region prior to reaching the stress at the yield point, but once in the above-theknee region, it requires only a small increment in load to produce a sufficiently large increase in deflection to reach the yield point. Thus the corresponding yield load may be used as an adequate approximation of the buckling load for a slender column as well. Hence the yield strength provides an adequate design bound for both massive and slender columns. It is also important to note that, in general, columns found in applications are sufficiently massive that the linear theory developed here is valid within the range of deflection that is of interest. Application of Eq. (30.1) to a simply supported imperfect column with constant properties over its length yields a modification of Eq. (30.4). Thus, d2y + k2y = k2 (e Y) dx2 (30.17)

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