46 A. Giordano and F.-P. Chiang From Eq. (4.3) we can obtain the y-curvature contour as follows: κy = ∂ ∂y ∂w ∂y . (4.5) From Eq. (4.2) or Eq. (4.3), we can obtain the twist curvature contour as follows: κxy = ∂ ∂y ∂w ∂x = ∂ ∂x ∂w ∂y . (4.6) Using the slope data deflection can be obtained via numerical integration. From Eq. (4.2) or Eq. (4.3), we can obtain the deflection contour as follows: w= ∂w ∂x dx = ∂w ∂y dy (4.7) Using the material properties of the plate the flexural rigidity, D, can be determined as follows: D= Et 3 12 1−ν2 (4.8) where Eis the Young’s modulus, t is the plate thickness, andν is the Poisson’s ratio. From the curvatures obtained in Eqs. (4.4, 4.5, and 4.6), and the flexural rigidity, Ddetermined in Eq. (4.8), the following relations exist between the moments per unit length Mx, My, and Mxy, and the plate curvatures [9]: Mx =−D κx +νκy My =−D κy +νκx Mxy =−D(1−ν)κxy (4.9) The fiber stresses of the plate due to the bending and twisting moments are given by: σx = 6Mx t2 σy = 6My t2 τxy = 6Mxy t2 (4.10) It is seen from Eqs. (4.9) and (4.10) that the stresses are related to the values of curvatures and twist, the second derivatives of plate deflection. Once the curvatures and the twist are experimentally determined, the stresses can be determined.
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