ABSTRACT
A generalized function can be used to represent loads on structures, the simplest example being a thin elastic string under tension (Section 2.1) with ”xed ends. If the transverse forces acting along the length of the string are represented by dižerentiable functions, the shear stress or force per unit length is an ordinary function (Section 2.2). If there are (Section 2.3) one or more concentrated forces, for example, a ”nite transverse force acting at a point, then (1) the shear stress is zero outside the point and in”nite at the point; and (2) the integral of the shear stress, namely the transverse force, has a ”nite jump as it passes through the point where the concentrated force is applied. ¡us the idealization of a concentrated force applied at a point leads to a transverse force (shear stress) that involves (Section 2.3) the generalized function Heaviside unit jump (Dirac unit impulse). ¡e string is de›ected under load, and its shape when a unit impulse loading is applied speci”es the linear (Section 2.3) [nonlinear (Section 2.6)] in›uence or Green function; the string has a triangular shape when a point load is applied both in the linear (Section 2.3) and nonlinear (Section 2.6) cases. ¡e dižerential equation specifying the shape of the string is nonlinear (linear) if the slope is not (is) small everywhere, for example, for the large (small) de›ection of a string under its own weight [Section 2.4 (Section 2.8)]; the large de›ection changes the parabolic shape of the small de›ection. In the linear case, the shape of the string under arbitrary loading can be calculated by the principle of superposition, using a convolution integral with the in›uence or Green function (Section 2.3); also, the principle of reciprocity holds, allowing interchange of the points of application of the concentrated load and of measurement of the de›ection. In contrast, the nonlinear Green function satis”es neither the principle of superposition nor the principle of reciprocity. ¡e types of loading considered include a transverse force (shear stress) that is (1) a linear (constant) function for the uniform loading of a homogeneous string in the linear (Section 2.4) and nonlinear (Section 2.7) cases, (2) a concentrated load represented by a generalized jump (impulse) function in the linear (Section 2.3) and nonlinear (Section 2.6) cases, and (3) multiple concentrated loads both in the linear (Section 2.5) and nonlinear (Example 10.5) cases. ¡e de›ection of an elastic string under its own weight corresponds to a uniform load in the linear case (Section 2.4) but not in the nonlinear case (Section 2.8). ¡e nonlinear de›ection of an elastic string under its own weight (Section 2.8) is larger than for the catenary, which corresponds to an inelastic string (Section 2.9); the inelastic string has constant length even for a nonlinear de›ection, whereas the elastic string (1) has constant length in the linear approximation (Sections 2.3 through 2.5) and (2) increases in length under a transverse load in the nonlinear case (Sections 2.6 through 2.8).
