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lecture19-20

# lecture19-20 - The component associated with the stiffness...

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Unformatted text preview: The component associated with the stiffness matrix 1 and ply coordinate h is called the extensional stiffness matrix, [A], as it holds the relation between in~plane force {N}and in—plane deformation That is, {N} = {Alb} where ’1 [A] z Z (hk ‘ hk—l) k=1 k Apparently, [A] has a size of 3 x 3. The existence of A16 and A26 implies that there are coupling effects between normal forces, NM and NW, and shear deformation, 3/1), , or shear force, ny, and normal deformations, 6H and 8”. B. Bending Stiffness Matrix A similar analysis can be extended to pure bending response of composite laminates. Recall the following relation between bending strain and curvature, 5 2 [(2 where a is the normal strain due to bending, K is the bending curvature and z is the thickness coordinate given in the following diagram. Expanding 8 to include all three strains in the two—dimensional domain and K to include bending curvatures in both x- and y-directions and twisting curvature in the x-y plane, it yields 8m Kxx {y} 2 5y), : {K}z 2 K», 2 m K, Similarly, by definition, the following relation holds Mxx 0x): M W = J. a”, zdz M W r”, where M’s are so-called resultant moments. Their units are of force. As an abbreviated form, the above equation can be expressed as {M } == [{0‘ }ZdZ With the substitution of the stress-strain relation, i.e. {a} = [film and the strain-curvature relation, i.e. {r}: {Kiz into the equation, it yields {M } = ]{y }zdz : ]{K}22dz Since composite laminates are made from stacking composite plies with different fib r orientations together and different composite plies have different stiffness matrices [Q], the integration in the above equation should be replaced by summation, i.e. I’l {Mkliﬁlkw=lzﬁlRM}%L) 3 k=l k 3 k=l k As mentioned earlier, all composite plies have identical deformation due to perfect bonding among them. Hence, {K} is constant for all composite plies and can be taken out of the summation, i.e. Vl {Mhlsza—aaa} 3 k:l k The component associated with the stiffness matrix ] and ply coordinate h is called the bending stiffness matrix, [D] , as it holds the relation between out-of-plane moment {M }and out—of—plane curvature {K}. That is, {M} = [DliKl where 1 n __ bkgzbhw—aa k=l k Apparently, [D] has a size of 3 x 3. The existence of D16 and D26 implies that there are coupling effects between bending moments, Mm and MW, and twisting curvature, K or xy ’ twisting moment, Mxy, and bending curvatures, Km and KW. C. Coupling Stiffness Matrix From examining the extensional and bending equations, {N} = [Alia/l {M } = [Dlilcl there exist possibilities of coupling effects between {N} and {K} and between {M }and {y} , i.e. {N} = [BM {M l = [Bliyl The matrix [B] in the above equations is called the coupling stiffness matrix because it holds relations between in—plane force {N}and out-of-plane curvature {K} and between out-of-plane moment {M }and in-plane deformation The coupling stiffness matrix is deﬁned as follows Bkéibhﬁ—MJ It should be noted that composite laminates with symmetric stacking sequences have zero [B] matrix. That is, no coupling effects between in—plane forces and out-of-plane curvatures and between out—of-plane moments and in—plane deformations. On the contrary, for nonzero [B], a 6 X 6 ABBD matrix should be sought, i.e. M: iii} More speciﬁcally, the relation between loading and deformation should be Nxx A11 A12 A16 BM 812 316 gm NW A12 A22 A26 B 12 B 22 B 26 an N xy 2 A16 A26 A66 316 B26 866 7w Mxx 811 BIZ 816 D11 D12 D16 Kxx M yy 8 12 B 22 B 26 D12 D22 D 26 yy Mxy *3 16 326 B 66 Dlé D26 D66 ny ...
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lecture19-20 - The component associated with the stiffness...

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