378 Mechanics of Composite Materials,Second Edition where U and U are the stiffness invariants given by Equation(2.132)and h is the thickness of the laminate.Also,find the in-plane engineering stiffness constants of the laminate. Solution From Equation(2.131a),for a general angle ply with angle 0, u=L U2 Cos20 Us Cos40. (5.9) For the kth ply of the quasi-isotropic laminate with an angle0 (Q)=U U2 Cos20g+Us Cos40x (5.10) where 0=5=8=贤-N1,s= 2π From Equation (4.28a), N A1= (5.11) k=1 where f=thickness of kth lamina. Because the thickness of the laminate is h and all laminae are of the same thickness, Nk=1,2,,N, k= (5.12) and,substituting Equation(5.10)in Equation(5.11), A=02u+u,cos29+山，cas46) k=】 (5.13) =hu,+u0∑cos29+u0∑cos40 2006 by Taylor Francis Group,LLC378 Mechanics of Composite Materials, Second Edition where U1 and U4 are the stiffness invariants given by Equation (2.132) and h is the thickness of the laminate. Also, find the in-plane engineering stiffness constants of the laminate. Solution From Equation (2.131a), for a general angle ply with angle θ, = U1 + U2 Cos2θ + U3 Cos4θ. (5.9) For the kth ply of the quasi-isotropic laminate with an angle θk, = U1 + U2 Cos2θk + U3 Cos4θk, (5.10) where From Equation (4.28a), , (5.11) where tk = thickness of kth lamina. Because the thickness of the laminate is h and all laminae are of the same thickness, (5.12) and, substituting Equation (5.10) in Equation (5.11), (5.13) Q11 ( ) Q11 k θ π θ π θ π θ π 12 1 θ π 2 1 = = …= … = − − = N N k N N N , ,, ,, kN N ( ) , . A tQk k k N 11 11 1 = = ∑ ( ) t h N k = = , , ,............, , k N 1 2 A h N UU U hU U k k k N 11 1 2 3 1 1 =+ + 2 4 = + = ∑( ) Cos Cos θ θ 2 3 1 1 2 4 h N U h N k k N k k N Cos Cos . θ θ + = = ∑ ∑ 1343_book.fm Page 378 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC