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Additional resources for Aircraft Structures 3E
As a solution to the stress analysis problem an Airy stress function q5 is proposed, where +=-- 120d3 + [5(x3- Z2x)(y d ) 2 ( y- 2d) - 3yx(y2 - d2)2] Show that q5 satisfies the internal compatibility conditions and obtain the distribution of stresses within the plate. Determine also the extent to which the static boundary conditions are satisfied. ax PX = -[5y(x2 - z2) - ioy3 ay = Txy = 20d3 E 4d3 ( + 6dZy] - 3yd2 - 2d3) -P [5(3x2- Z2)(y2- d2) - 5y4 + 6y2d2- $1 40d3 The boundary stress function values of T~~do not agree with the assumed constant equilibrating shears at x = 0 and 1.
31) Turning our attention now to the triangle C‘F’E’ we have (C‘E’)? 32) in which C’E’ = CE( 1 + E,,) + F’E’ = FE(l + E,+,/z) C’F‘ = CF( 1 E,) Substituting for C’E’, C’F’ and F’E’ in Eq. 33) All the strains are assumed to be small so that their squares and higher powers may be ignored. Further, siny M y and Eq. 33) becomes + ) (CF)’(1 (CE)2(1 2 ~ y = + 2 ~ n+) (FE)2(1+ 2~,+,/2) - 2(CF)(FE)y From Fig. 1. 9) by replacing a, by E,, u, by E ~ uy yxy/2 and T by y/2. 34) regarding E, and y/2. Thus, at a point in a deformable body, there are two mutually perpendicular planes on which the shear strain y is zero and normal to which the direct strain is a maximum or minimum.
A more detailed examination of this effect is instructive. The shear strain at any point in the beam is given by Eq. (vi) P 8ZG yxy = - -(b2 - 4y2) and is obviously independent of x . Therefore at all points on the neutral plane the shear strain is constant and equal to y Pb2 8IG =-- xy which amounts to a rotation of the neutral plane as shown in Fig. 6. The deflection of the neutral plane due to this shear strain at any section of the beam is therefore equal to Pb2 8IG -(I-X) Fig. 6 Rotation of neutral plane due to shear in end-loaded cantilever.
Aircraft Structures 3E by Megson