## 342 Elastic Lateral Buckling Of Beams

Bending of the beam shown in Fig. 3.90a produces compressive stresses within the upper portion of the beam cross section and tensile stresses in the lower portion. Similar to the behavior of a column (Art. 3.41), a beam, although the compressive stresses may be well within the elastic range, can undergo lateral buckling failure. Unlike a column, however, the beam is also subjected to tension, which tends to restrain the member from lateral translation. Hence, when lateral buckling of the beam occurs, it is through a combination of twisting and out-of-plane bending (Fig. 3.90b).

For a simply supported beam of rectangular cross section subjected to uniform bending, buckling occurs at the critical bending moment

where L = unbraced length of the member E = modulus of elasticity Iy = moment of inertial about minor axis G = shear modulus of elasticity J = torsional constant

As indicted in Eq. (3.170), the critical moment is proportional to both the lateral bending stiffness EIy/L and the torsional stiffness of the member GJ/L.

For the case of an open section, such as a wide-flange or I-beam section, warping rigidity can provide additional torsional stiffness. Buckling of a simply supported beam of open cross section subjected to uniform bending occurs at the critical bending moment

where Cw is the warping constant, a function of cross-sectional shape and dimensions (see Fig. 3.89).

In Eq. (3.170) and (3.171), the distribution of bending moment is assumed to be uniform. For the case of a nonuniform bending-moment gradient, buckling often occurs at a larger critical moment. Approximation of this critical bending moment M'cr may be obtained by multiplying Mcr given by Eq. (3.170) or (3.171) by an amplification factor:

b 2.5Mmax + 3Ma + 4Mb + 3Mc Mmax = absolute value of maximum moment in the unbraced beam segment Ma = absolute value of moment at quarter point of the unbraced beam segment Mb = absolute value of moment at centerline of the unbraced beam segment MC = absolute value of moment at three-quarter point of the unbraced beam segment

FIGURE 3.90 (a) Simple beam subjected to equal end moments. (b) Elastic lateral buckling of the beam.

FIGURE 3.90 (a) Simple beam subjected to equal end moments. (b) Elastic lateral buckling of the beam.

Cb equals 1.0 for unbraced cantilevers and for members where the moment within a significant portion of the unbraced segment is greater than or equal to the larger of the segment end moments.

(S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, and F. Bleich, Buckling Strength of Metal Structures, McGraw-Hill, Inc., New York; T. V. Galambos, Guide to Stability of Design of Metal Structures, John Wiley & SOns, Inc., New York; W. McGuire, Steel Structures, Prentice-Hall, Inc., Englewood Cliffs, N.J.; Load and Resistance Factor Design Specification for Structural Steel Buildings, American Institute of Steel Construction, Chicago, Ill.)

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