Can you do problem 2 and 4 in the photo attached. Example 2.2 and 2.3 is given in pdf from page 8
Slide 1 11/09/2018 1 Theoretical Determination of Properties: 3. Laminate Level • A multidirectional laminate is constructed from number of unidirectional plies • The laminas (plies) are stacked at various on- and off-axis angles relative to the global 1,2,6 axis • Stacking sequence is used to define the layup • Stacking sequence shows the individual plies of the laminate, from bottom to top in ordered sequence of angles such as [0/90/+45/-45]. • Usually all plies have the same thickness and same materials, if the thickness or the material is different, then a modification to the stacking seq. is required 1 Theoretical Determination of Properties: 3. Laminate Level • A multidirectional laminate may be orthotropic or anisotropic in its plane or out of its plane • It may has coupling between in-plane and out-of plane responses. • In plane response is known as extensional response and out of plane response is known as flexural response. 2 11/09/2018 2 Theoretical Determination of Properties: 3. Laminate Level • Classical lamination theory (CLT) is generally used as first approximation to describe the constitutive relations of a thin laminate • According to CLT, in-plane response is described by 3 extensional stress resultants and out of plane response is described by three flexural stress resultants • Correspondingly, there are six generalized strain resultants : the mid-plane strains, , and plate curvature 621 ,, NNN 621 ,, MMM 0 6 0 2 0 1 ,, 621 ,, 3 Theoretical Determination of Properties: 3. Laminate Level • The general constitutive relation for multidirectional laminate is given as: (2-54) The stiffness matrix is called [A-B-D] matrix 6 2 1 0 6 0 2 0 1 666261666261 262221262221 161211161211 666261666261 262221262221 161211161211 6 2 1 6 2 1 DDDBBB DDDBBB DDDBBB BBBAAA BBBAAA BBBAAA M M M N N N 4 11/09/2018 3 Theoretical Determination of Properties: 3. Laminate Level • defines the extensional response • defines the flexural response, • defines the extension-bending coupled response • The entire [A-B-D] matrix is symmetric • [A], [B], and [D] sub-matrices are symmetric • Then, [A-B-D] has 18 independent terms • [A-B-D] matrix can be obtained by summation of the stiffness coefficients of the individual on- and off- plies: (2-55) ij A ij D ij B n k k k ij n k kk k ijij hQzzQA 11 1 5 Theoretical Determination of Properties: 3. Laminate Level (2-56) (2-57) Where, is the distance from mid-plane to the top ply k, the distance to the middle of ply k, the thickness of ply k, total number of plies in the laminate is positive for a ply above the mid-plane and negative for a ply below the mid-plane n k kk k ij n k kk k ijij hzQzzQB 11 2 1 2 2 1 n k k kk k ij n k kk k ijij h zhQzzQD 1 3 2 1 3 1 3 123 1 k z k z k h n z 6 11/09/2018 4 Theoretical Determination of Properties: 3. Laminate Level nk ik 2k 1k 2 hh k 1 zz k 0, zmidplane 1 zz k 7 Theoretical Determination of Properties: 3. Laminate Level A very common laminate layup is symmetric about mid- plane In this case, B sub-matrix is zero Only in very special cases, unsymmetric laminate may used Unsymmetric laminates have extensional-bending coupling that can affect the response of thin laminates (especially those used for strengthening applications) In symmetric laminate the extensional and bending response and un-coupled (although they can have bending-twisting coupling) and the individual [A] and [D] submatrices can be inverted separately to give extensional and bending compliance matrices, known as [a] and [d] 8 11/09/2018 5 Theoretical Determination of Properties: 3. Laminate Level (2-58) (2-59) • Effective in-plane engineering constants can then be defined for symmetric laminate of thickness by defining an average in- plane stress in the laminate, as follows: (2-60) • The compliance relationship then becomes: (2-61) or, (2-62) 6,2,1,1 jiAa ijij 6,2,1,1 jiDd ijij h 6,2,1, ji h N i i 6,2,1,0 jiah jiji 6 2 1 0 6 0 2 0 62 0 1 0 61 0 6 0 26 0 2 0 1 0 21 0 6 0 16 0 2 0 12 0 1 0 6 0 2 0 1 1 1 1