Elasto-Plastic Bending of Beams on Elastic Foundations

Abstract

Two methods for the determination of elasto-plastic deformations of elastically supported beams having a bilinear stressstrain curve are presented and compared. One method involves the numerical solution of the fourth-order nonlinear differential equation associated with the engineering theory of beam bending while the other linearizes the problem by replacing the nonlinear moment-curvature diagram with its linear asymptotes. Results obtained through the two methods are in excellent agreement. The linearized method is recommended for investigation of the initial phases of bending of elastically supported beams in which onty a few plastic regions appear. Neither method has any particular advantages when the beam has many plastic regions— i.e., both methods involve a tremendous amount of computation. Computed deflections for finite elastically supported beams are in good agreement with experimental results; differences of five per cent are consistent in the elastic and in most of the plastic regions. SYMBOLS b = width of beam at any layer in the cross section; maximum width of beam E = Young's modulus of beam material E = slope of linear strain hardening portion of stressstrain curve h = half-depth of beam cross section IQ = moment of inertia of beam cross section k = foundation modulus L = length of finite beam M = bending moment at any beam station Mp = maximum possible bending moment in perfectly plastic beam M* = bending moment causing yielding of outermost beam fibers (M* = <r*I0/h) P = applied concentrated load P* = concentrated load for which yielding begins at outermost beam fibers p * = MPP/M* q = distributed load on beam q = nondimensional distributed load (q = q/M*\) q = nondimensional distributed load (q = q/Mp ) t = nondimensional bending moment ratio (t = M/M*) t = nondimensional bending moment ratio (I = M/Mp) V = shear force w = beam deflection w* = beam deflection at point of application of concentrated load when yielding begins w* = Mpw*/M* x = distance along beam y = distance from neutral axis to any layer in beam cross section 8 = Xm ~ Xp X Xm X Xp Xp Presented at the Structures Session, Twenty-Third Annual Meeting, IAS, New York, January 24-27, 1955. t The present paper is essentially a condensation of the author's dissertation for the degree of Doctor of Philosophy at Stanford University, prepared under contract N60NR-251 Task Order 11 (NR-064-240) for the Office of Naval Research. t Research Engineer, Structural Research Group. = strain = yield strain = (k/AEIQf u = beam radius of curvature = stress = yield stress = nondimensional distance along beam (% ~ Xs) = nondimensional length of finite beam (£m = XL) = nondimensional distance along beam (x = \x) = nondimensional length of first plastic region = nondimensional length of first plastic region [xP = O^MXP] \p = ratio of depths of elastic core and cross section sgn ( ) = + 1 if quantity in parentheses is positive, —1 if quantity is negative X = (k/4EI0y /i