The expression for deflection surface of plate which satisfies the boundary conditions Eq. (27-389) and Eq. (27-388b) Substituting Eq. (27-390) in Eq. (27-388b) and solving for C Equation (27-390) for w becomes The expression for M x , M y and M xy The maximum deflection and bending moments, which occur at midpoint of plate The maximum deflection and bending moments for a square plate The shearing forces from Eqs. (27-368) w ¼ C sin x a sin y b ð27-390Þ C ¼ q 0 4 D 1 a 2 þ 1 b 2 2 ð27-390aÞ w ¼ q 0 4 D 1 a 2 þ 1 b 2 sin x a sin y b ð27-390bÞ M x ¼ q 0 2 1 a 2 þ 1 b 2 2 1 a 2 þ v b 2 sin x a sin y b ð27-391aÞ M y ¼ q 0 2 1 a 2 þ 1 b 2 2 v a 2 þ 1 b 2 sin x a sin y b ð27-391bÞ M xy ¼ q 0 ð1 À vÞ 2 1 a 2 þ 1 b 2 2 ab sin x a sin y b ð27-391cÞ w max ¼ q 0 4 D 1 a 2 þ 1 b 2 2 ð27-392aÞ M x max ¼ q 0 4 D 1 a 2 þ 1 b 2 2 1 a 2 þ v b 2 ð27-392bÞ M y max ¼ q 0 2 1 a 2 þ 1 b 2 2 v a 2 þ 1 b 2 ð27-392cÞ w max ¼ q 0 a 4 4 4 D ; M x max ¼ M y max ¼ ð1 þ vÞq 0 a 2 4 2 ð27-393Þ Q x ¼ q 0 a 1 a 2 þ 1 b 2 cos x a sin y b ð27-394aÞ Q y ¼ q 0 b 1 a 2 þ 1 b 2 sin x a cos y b ð27-394bÞ Particular Formula 27.82 CHAPTER TWENTY-SEVEN Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. APPLIED ELASTICITY The reactive forces at the support edges at x ¼ a and y ¼ b respectively The resultant reaction concentrated at the corners of the plate The total pressure on all four edges of plate The four corners reactions, which are equal due to symmetry The maximum bending stress if a > b is due to M y which is greater than M x Using Eq. (27-395d), the expression for maximum shear stress which is at the middle of the longer side of the plate V x ¼ Q x À @M xy @y x ¼a ð27-395aÞ V x ¼À q 0 a 1 a 2 þ 1 b 2 2 1 a 2 þ 2 À v b 2 sin y b ð27-395bÞ V y ¼ Q y À @M xy @x y ¼b ð27-395cÞ V y ¼À q 0 b 1 a 2 þ 1 b 2 2 1 b 2 þ 2 À v a 2 sin x a ð27-395dÞ R ¼ 2ðM xy Þ x ¼a y ¼b ¼ 2q 0 ð1 À vÞ 2 ab 1 a 2 þ 1 b 2 2 ð27-396Þ 2 ð b 0 v x dy þ 2 ð a 0 v y dx ¼ 4q 0 ab 4 þ 8q 0 ð1 À vÞ 2 ab 1 a 2 þ 1 b 2 2 ð27-397Þ ÁR ¼ 8q 0 ð1 À vÞ 2 ab 1 a 2 þ 1 b 2 2 which is the second term on the right hand side of Eq. (27-397) y max ¼ 6M y max h 2 ¼ 6q 0 2 h 2 1 a 2 þ 1 b 2 2 v a 2 þ 1 b 2 ð27-398Þ ð yz Þ max ¼ 3q 0 2bh 1 a 2 þ 1 b 2 2 1 b 2 þ 2 À v a 2 ð27-399Þ Particular Formula APPLIED ELASTICITY 27.83 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. APPLIED ELASTICITY . vÞq 0 a 2 4 2 ð27-393Þ Q x ¼ q 0 a 1 a 2 þ 1 b 2 cos x a sin y b ð27-394aÞ Q y ¼ q 0 b 1 a 2 þ 1 b 2 sin x a cos y b ð27-394bÞ Particular Formula 27.82 CHAPTER TWENTY-SEVEN Downloaded from Digital Engineering Library @ McGraw-Hill. max h 2 ¼ 6q 0 2 h 2 1 a 2 þ 1 b 2 2 v a 2 þ 1 b 2 ð27-398Þ ð yz Þ max ¼ 3q 0 2bh 1 a 2 þ 1 b 2 2 1 b 2 þ 2 À v a 2 ð27-399Þ Particular Formula APPLIED ELASTICITY 27.83 Downloaded from Digital Engineering Library @ McGraw-Hill