ABSTRACT
A bar, like a string, is an elastic body whose longitudinal dimension or length is much larger than the cross section; in the case of the string (bar), the cross section is (is not) negligible, and there is no (there is) bending stižness. ¡us, a string cannot support a bending moment; a bending moment causes a straight bar to become curved. ¡e curvature of a string is due to a shear stress, and thus, the displacement satis”es a second-order dižerential equation (Chapter 2); for a bar (Chapter 4), the bending moment is proportional to the curvature (Section 4.1), leading to a second-order dižerential equation that becomes of third order for the transverse force and of fourth order for the shear stress. As for a string (Chapter 2), the minimum of the elastic energy speci”es (Section 4.1) the balance equation and boundary conditions. Since the displacement of a string (bar) satis”es a second (fourth)-order dižerential equation, there are two (four) boundary conditions, that is, one (two) at each end, indicating the coordinate of the suspension point (the method of attachment). A bar may be clamped, pin-joined, or have a sliding support or it may have a free end (Section 4.2); there are four possible static combinations of these three attachment conditions, each associated with a dižerent set of boundary conditions. Each set of boundary conditions leads to a distinct Green or in›uence function that speci”es the shape of the elastica or neutral line for a bar subject to a concentrated transverse force (Section 4.4). ¡e in›uence function can then be used with the superposition principle to specify the shape of the bar under arbitrary loads, for example, its own weight (Section 4.3) or a concentrated moment (Section 4.5). ¡e superposition principle applies only for linear bending, that is, with a small slope; this is a particular limit of strong bending, when the general nonlinear balance equation simpli”es to linear. ¡e linear (nonlinear) bending of a bar [Sections 4.3 through 4.5 (Sections 4.7 through 4.9)] can be considered for similar loads (Section 4.1) and boundary conditions (Section 4.2), for example, (1) a concentrated transverse force [Section 4.4 (Sections 4.8 and 4.9)] specifying the linear (nonlinear) response function; (2) one or several concentrated moments [Section 4.5 (Section 4.7)]; and (3) its own weight [Section 4.3 (Section 4.8)]. A beam is a bar subject to a bending moment plus an axial traction (like a string) or a compression (Section 4.1); if a bar has a free end, the bending causes no change in length and no compression or traction. If neither end is free, a beam under strong bending cannot change the distance between the supports, and the extension is associated with axial tractions (Section 4.6). ¡us, to the transverse loads may be added a longitudinal tension either (1) as an externally applied traction or compression or (2) as a consequence of the bar being pin-joined or clamped at both ends and its extension under bending giving rise to a longitudinal traction. If the de›ection of the neutral line or elastica is small (large) compared with the thickness of the cross section, the one-dimensional elastic body behaves like a bar (string); if they are comparable, the two ežects are superimposed, which is the case of a beam. ¡ere are two basic cases of elastic instability (Section 4.9): (1) the linear buckling of a bar under an axial load and (2) the nonlinear collapse of a bar sliding between supports due to a transverse load. Two more cases of elastic instability concern a cantilever bar that is clamped at one end, with the other supported on a linear (rotary) spring [Example 10.10 (10.11)] placed so as to increase the de›ection.
