06_PlateTheory_02_MomentCurvature

06_PlateTheory_02_MomentCurvature - Section 6.2 6.2 The...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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 = (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 ( zz 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 / = . One finds that x y initial position w
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Section 6.2 Solid Mechanics Part II Kelly 126 () [] 2 / 3 2 2 2 / 1 / x x x + = ω κ (6.2.2) Also, the radius of curvature x R , Fig. 6.2.2, is the reciprocal of the curvature, x x R / 1 = . Fig. 6.2.2: Angle and arc-length used in the definition of curvature As with the beam, when the slope is small , one can take x w = / tan ψ and x ds d / / and Eqn. 6.2.2 reduces to (and similarly for the curvature in the y direction) 2 2 2 2 1 , 1 y w R x w R y y x x = = = = (6.2.3) This important assumption of small curvature, or equivalently of assuming that the slope 1 / << x , means that the theory to be developed will be valid when the deflections are small compared to the overall dimensions of the plate. The curvatures 6.2.3 can be interpreted as in Fig. 6.2.3, as the unit increase in slope along the x and y directions. Figure 6.2.3: Physical meaning of the curvatures y w x y AB C D A B y y y y Δ + 2 2 A x x x x Δ + 2 2 x C ww y Δ x Δ x w s x R
Background image of page 2
Section 6.2 Solid Mechanics Part II Kelly 127 Twist Not only does a plate curve up or down, it can also twist (see Fig. 6.1.3). The twist is defined analogously to the curvature and is denoted by xy T / 1: y x w T xy = 2 1 (6.2.4) The physical meaning of the twist is illustrated in Fig. 6.2.4. Figure 6.2.4: Physical meaning of the twist Principal Curvatures Consider the two Cartesian coordinate systems shown in Fig. 6.2.5, the second ( n t ) obtained from the first ( y x ) by a positive rotation θ . The partial derivatives arising in the curvature expressions can be expressed in terms of derivatives with respect to t
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 14

06_PlateTheory_02_MomentCurvature - Section 6.2 6.2 The...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online