The End Problem of Rectangular Strips

Abstract

Abstract J. Fadle was first to derive a solution for the stresses in a rectangular strip subject on one of its narrow edges to a self-equilibrating system of stresses; i.e., stresses which have zero resultant force and zero resultant moment. Fadle’s solution is rather unwieldy because of the use of complex (biharmonic) eigenfunctions. These eigenfunctions are associated with the roots of the equation sin 2γ ± 2γ = 0. The first (nonvanishing) root, as determined to seven decimals by Mittelman and Hillman, is γ2 = 2.1061961 + 1.1253643i. In this paper we derive a variational solution of the same problem in the form φ(x, y) = Σcnφn where, in first approximation, φn = fn(y)gn(x). The series Σcnfn(y) is obtained by expansion of the boundary values φ(0, y) of the Airy function φ into a complete set of self-equilibrating orthogonal polynomials fn(y); the functions gn(x) are then determined from the Euler-Lagrange equations of the associated variational problem. This procedure has the advantage of staying entirely in the real domain. The first approximation corresponds to the physical concept that the longitudinal fibers of the strip are beams on elastic foundation (the adjacent fibers furnish the elastic foundation) subject to end thrusts and lateral loads. Incidental to this idea there are provided quantitative formulas which relate depth of penetration of an applied traction to its shape (number of its wiggles). The eigenvalues γn, the real parts of which furnish the extinction coefficients, are obtained from a quadratic equation in γ2. One finds that the lowest (nonvanishing) eigenvalue is γ2 = 2.075 + 1.143i. In higher approximations coupling effects between derivatives of the orthogonal modes fngn are also taken into account. The jth approximation to a mode φn is obtained by evaluating a 2j − 1-rowed determinantal equation involving the functions fn−j+1 gn−j+1 to fn+j−1 gn+j−1. High modes are calculated with the same facility and accuracy as is the fundamental except that the fundamental mode φ2 not being preceded by other modes, requires, in jth approximation, only a j-rowed determinantal equation. One thus finds that in third approximation γ2 is the root of a 6th degree equation in γ2, and has the value γ2 = 2.1061964 + 1.1253644i.