06_PlateTheory_02_MomentCurvature - Section 6.2 6.2 The...

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Section 6.2 Solid Mechanics Part II Kelly 125 6.2 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 y positive up, R y xx / = ε , where R is the radius of curvature , R positive when the beam bends “up” (see Part I, Eqn. 4.6.16). In terms of the curvature R x v / 1 / 2 2 = , where v is the deflection (see Part I, Eqn. 4.6.35), one has 2 2 x v y xx = 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 xx σ and yy σ ( zz σ is 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 y x plane. When deformed, the mid-surface occupies the surface ) , ( y x w w = and w is the elevation above the y x plane, Fig. 6.2.1. Fig. 6.2.1: Deformed Plate The slopes of the plate along the x and y directions are x w / and y w / . Curvature Recall from Part I, §4.6.10, that the curvature in the x direction, x κ , is the rate of change of the slope angle ψ with respect to arc length s , Fig. 6.2.2, ds d x / ψ κ = . One finds that x y initial position w
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