By T.H.G. Megson
Aircraft buildings for Engineering scholars is the top self contained airplane buildings path textual content. It covers all primary matters, together with elasticity, structural research, airworthiness and aeroelasticity. Now in its fifth version, the writer has revised and up to date the textual content all through and extra new examples and routines utilizing Matlab(c). extra labored examples make the textual content much more obtainable by means of displaying program of recommendations to airframe constructions. contains a options handbook to be had to all adopting teachers.
* New labored examples in the course of the textual content relief realizing and relate suggestions to genuine global purposes * Matlab examples and workouts extra all through to help use of computational instruments in research and design
* an in depth airplane layout venture case learn exhibits the applying of the most important recommendations within the publication * extra finish of bankruptcy routines, with an accompanying Solutions handbook (for teachers in simple terms) at http://textbooks.elsevier.com
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The appropriate equilibrium conditions for plane stress are given by Eqs. G. Megson. Published by Elsevier Ltd. All rights reserved. 47 48 CHAPTER 2 Two-dimensional problems in elasticity @sx @txy þ þX ¼0 @x @y @sy @tyx þ þY ¼0 @y @x and the required stress–strain relationships obtained from Eqs. 47): 1 ðsx À nsy Þ E 1 ey ¼ ðsy À nsx Þ E ex ¼ gxy ¼ 2ð1 þ nÞ txy E We find that, although ez exists, Eqs. 26) are identically satisfied, leaving Eq. 21) as the required compatibility condition. Substitution in Eq.
3 54 CHAPTER 2 Two-dimensional problems in elasticity Substituting in Eq. 9) gives @4f @4f @4f þ 2 2 2 þ 4 ¼ 2 Â 60Dy À 120Dy ¼ 0 4 @x @x @y @y The biharmonic equation is therefore satisfied and the stress function is valid. From Fig. 3, sy ¼ 0 at y ¼ h, so that, from Eq. (i), 2A þ 2BH þ 10Dh3 ¼ 0 (iv) Also, from Fig. 3, vy ¼ –q at y ¼ –h, so that, from Eq. (i), 2A À 2BH À 10Dh3 ¼ Àq (v) Again, from Fig. 3, txy ¼ 0 at y ¼ Æ h, giving, from Eq. (iii), 2Bx þ 30Dxh2 ¼ 0 so that 2B þ 30Dh2 ¼ 0 (vi) At x ¼ 0, no resultant moment is applied to the beam; that is, ðh ðh sx y dy ¼ ð6Cy2 À 20Dy4 Þ dy ¼ 0 Mx¼0 ¼ Àh Àh so that Â Ãh Mx¼0 ¼ 2Cy3 À 4Dy5 Àh ¼ 0 or C À 2Dh2 ¼ 0 (vii) Subtracting Eq.
8. A circle of strain, analogous to that shown in Fig. 13. The horizontal extremities of the circle represent the principal strains, the radius of the circle, half the maximum shear strain, and so on. 15 STRESS–STRAIN RELATIONSHIPS In the preceding sections, we developed, for a three-dimensional deformable body, three equations of equilibrium (Eqs. 5)) and six strain-displacement relationships (Eqs. 20)). From the latter, we eliminated displacements, thereby deriving six auxiliary equations relating strains.