Section 6.2 Solid Mechanics Part II Kelly 1256.2 The Moment-Curvature Equations 6.2.1 From Beam Theory to Plate Theory In the beam theory, based on the assumptions of plane sections remaining plane and that one can neglect the transverse strain, the strain varies linearly through the thickness. In the notation of the beam, with ypositive up, Ryxx/−=ε, where Ris the radius of curvature, Rpositive when the beam bends “up” (see Part I, Eqn. 4.6.16). In terms of the curvatureRxv/1/22=∂∂, where vis the deflection (see Part I, Eqn. 4.6.35), one has 22xvyxx∂∂−=ε(6.2.1) The beam theory assumptions are essentially the same for the plate, leading to strains which are proportional to distance from the neutral (mid-plane) surface, z, and expressions similar to 6.2.1. This leads again to linearly varying stresses xxσand yyσ(zzσis also taken to be zero, as in the beam theory). 6.2.2 Curvature and Twist The plate is initially undeformed and flat with the mid-surface lying in the yx−plane. When deformed, the mid-surface occupies the surface ),(yxww=and wis the elevation above the yx−plane, Fig. 6.2.1. Fig. 6.2.1: Deformed PlateThe slopes of the plate along the xand ydirections are xw∂∂/and yw∂∂/. Curvature Recall from Part I, §4.6.10, that the curvature in the xdirection, xκ, is the rate of change of the slope angle ψwith respect to arc length s, Fig. 6.2.2, dsdx/ψκ=. One finds that xy••initial position w
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