Strength of Very Slender Beams

The response of a slender beam to lateral loads and twisting couples is affected by the presence of bending moments in the plane of major stiffness in the same manner as the bending of beams may be influenced by the presence of axial forces. In addition, if the major bending moments are statically indeterminate and if the beam is sufficiently slender to admit relatively large lateral deformations, these deformations may, in turn, affect the distribution of the principal bending moments. The resulting nonlinear theory is the subject of the paper. After establishing the basic equations, it is demonstrated that the inclusion of nonlinear terms in the strain-displacement relationships corresponds to a stiffening of the structure as compared with the familiar linear theory. The redistributed major bending moments and reactions satisfy a minimum principle that represents an extension of the classical Castigliano theorem. It is demonstrated further that for increasing lateral loads and torsional moments a limiting major bending-moment distribution is approached asymptotically. For certain singular cases the corresponding equilibrium configuration may not be unique. In this case, the possibility of a “snap-through” (Durchschlag) phenomenon arises. The theory presented herein is corroborated experimentally with a fair degree of accuracy. Elastic behavior is assumed throughout.