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Prob. 15.5 A 60 o/ 0 o/ 60 o layup gives an example of what are called "quasi-isotropic" laminates, having equal stiffnesses in the x and y directions, regardless of the laminate orientation. Verify that this is so for two laminate

  1. Prob. 15.5 A 60 o/ 0 o/ 60 o layup gives an example of what are called "quasi-isotropic" laminates, having equal stiffnesses in the x and y directions, regardless of the laminate orientation. Verify that this is so for two laminate orientations, one having the middle ply fibers oriented along the x axis and the other with the middle ply fibers oriented at 30 o from the x axis. Examine the B submatrices (Eqn. 15.22) for these cases, and explain how the laminates do not act as isotropic materials. First laminate: 1> plate assign properties for lamina type 1... enter modulus in fiber direction... (enter -1 to stop): 230e9 enter modulus in transverse direction: 6.6e9 enter principal Poisson ratio: .25 enter shear modulus: 4.8e9 enter ply thickness: .13e-3 assign properties for lamina type 2... enter modulus in fiber direction... (enter -1 to stop): -1 define layup sequence, starting at bottom... (use negative material set number to stop) enter material set number for ply number 1: 1 enter ply angle: 60 <=== 60/0/-60 layup enter material set number for ply number 2: 1 enter ply angle: 0 enter material set number for ply number 3: 1 enter ply angle: -60 enter material set number for ply number 4: -1 laminate stiffness matrix: 0.3576E+08 0.1110E+08 0.0000E+00 0.0000D+00 0.0000D+00 -0.8528D+03 0.1110E+08 0.3576E+08 0.0000E+00 0.0000D+00 0.0000D+00 -0.2423D+04 0.0000E+00 0.0000E+00 0.1233E+08 -0.8528D+03 -0.2423D+04 0.0000D+00 0.0000E+00 0.0000E+00 -0.8528E+03 0.1485D+00 0.1996D+00 0.0000D+00 0.0000E+00 0.0000E+00 -0.2423E+04 0.1996D+00 0.6402D+00 0.0000D+00 -0.8528E+03 -0.2423E+04 0.0000E+00 0.0000D+00 0.0000D+00 0.2152D+00 laminate compliance matrix: 0.3114E-07 -0.5507E-08 0.0000E+00 0.0000D+00 0.0000D+00 0.6140D-04 -0.5507E-08 0.1188E-06 0.0000E+00 0.0000D+00 0.0000D+00 0.1316D-02 0.0000E+00 0.0000E+00 0.3276E-06 0.3699D-03 0.1125D-02 0.0000D+00 0.0000E+00 0.0000E+00 0.3699E-03 0.1201D+02 -0.2345D+01 0.0000D+00 0.0000E+00 0.0000E+00 0.1125E-02 -0.2345D+01 0.6548D+01 0.0000D+00 0.6140E-04 0.1316E-02 0.0000E+00 0.0000D+00 0.0000D+00 0.1970D+02 Second laminate: 2> plate assign properties for lamina type 1... enter modulus in fiber direction... (enter -1 to stop): 230e9 enter modulus in transverse direction: 6.6e9

  2. enter principal Poisson ratio: .25 enter shear modulus: 4.8e9 enter ply thickness: .13e-3 assign properties for lamina type 2... enter modulus in fiber direction... (enter -1 to stop): -1 define layup sequence, starting at bottom... (use negative material set number to stop) enter material set number for ply number 1: 1 enter ply angle: 90 <=== 90/30/-30 layup enter material set number for ply number 2: 1 enter ply angle: 30 enter material set number for ply number 3: 1 enter ply angle: -30 enter material set number for ply number 4: -1 laminate stiffness matrix: 0.3576E+08 0.1110E+08 0.1000E+01 0.2157D+04 0.6798D+03 -0.1211D+04 0.1110E+08 0.3576E+08 0.3600E+02 0.6798D+03 -0.3516D+04 -0.4264D+03 0.1000E+01 0.3600E+02 0.1233E+08 -0.1211D+04 -0.4264D+03 0.6798D+03 0.2157E+04 0.6798E+03 -0.1211E+04 0.3598D+00 0.1113D+00 -0.1575D+00 0.6798E+03 -0.3516E+04 -0.4264E+03 0.1113D+00 0.6057D+00 -0.5543D-01 -0.1211E+04 -0.4264E+03 0.6798E+03 -0.1575D+00 -0.5543D-01 0.1268D+00 laminate compliance matrix: 0.8431E-07 -0.3676E-07 -0.7405E-07 -0.4042D-03 -0.2429D-03 0.4705D-03 -0.3676E-07 0.1281E-06 -0.1869E-08 -0.1921D-03 0.8391D-03 0.2180D-03 -0.7405E-07 -0.1869E-08 0.2026E-06 0.7496D-03 -0.2410D-05 -0.8699D-03 -0.4042E-03 -0.1921E-03 0.7496E-03 0.9979D+01 -0.1680D+01 0.3132D+01 -0.2429E-03 0.8391E-03 -0.2410E-05 -0.1680D+01 0.7249D+01 0.1597D+01 0.4705E-03 0.2180E-03 -0.8699E-03 0.3132D+01 0.1597D+01 0.2236D+02 Note that the 1,1 and 2,2 elements of the stiffness matrix are the same, and also the same for the second laminate oriented 30 o from the first. This is an apparent isotropy: stiffness the same in all directions (though only verified here for four directions). Neither laminate is symmetric, so there are nonzero coupling terms in the B submatrices, which is not expected in isotropic materials. The appearance of isotropy is not evident in the compliance matrix, where the 1,1 and 2,2 elements are different; this is a consequence of the extension-curvature coupling that occurs in the nonsymmetric laminate. If the laminate is made quasi-isotropic and symmetric, as in the 60/0/-60/-60/0/60 layup below, the appearance of isotropy occurs in both the stiffness and compliance matrices. 1> plate assign properties for lamina type 1... enter modulus in fiber direction... (enter -1 to stop): 230e9 enter modulus in transverse direction: 6.6e9 enter principal Poisson ratio: .25 enter shear modulus: 4.8e9 enter ply thickness: .13e-3 assign properties for lamina type 2... enter modulus in fiber direction... (enter -1 to stop): -1 define layup sequence, starting at bottom... (use negative material set number to stop) enter material set number for ply number 1: 1 enter ply angle: 60

  3. enter material set number for ply number 2: 1 enter ply angle: 0 enter material set number for ply number 3: 1 enter ply angle: -60 enter material set number for ply number 4: 1 enter ply angle: -60 enter material set number for ply number 5: 1 enter ply angle: 0 enter material set number for ply number 6: 1 enter ply angle: 60 enter material set number for ply number 7: -1 laminate stiffness matrix: 0.7152E+08 0.2221E+08 0.0000E+00 0.2441D-03 -0.1221D-03 0.0000D+00 0.2221E+08 0.7152E+08 0.0000E+00 0.0000D+00 0.0000D+00 0.2441D-03 0.0000E+00 0.0000E+00 0.2466E+08 0.0000D+00 0.0000D+00 0.1221D-03 0.2441E-03 -0.1221E-03 0.0000E+00 0.3017D+01 0.1244D+01 0.6652D+00 0.0000E+00 0.0000E+00 0.2441E-03 0.1244D+01 0.4000D+01 0.1890D+01 0.0000E+00 0.0000E+00 0.1221E-03 0.6652D+00 0.1890D+01 0.1368D+01 laminate compliance matrix: 0.1547E-07 -0.4804E-08 -0.1010E-22 -0.1685D-11 0.5872D-12 0.8656D-12 -0.4804E-08 0.1547E-07 0.5762E-23 0.7364D-12 0.3160D-11 -0.7484D-11 -0.1349E-23 0.9467E-24 0.4055E-07 0.3092D-12 0.4847D-11 -0.1046D-10 -0.1667E-11 0.1170E-11 0.1193E-11 0.3821D+00 -0.8927D-01 -0.6247D-01 0.3895E-12 -0.2733E-12 -0.2482E-11 -0.8927D-01 0.7403D+00 -0.9792D+00 0.2726E-12 -0.1913E-12 -0.7698E-12 -0.6247D-01 -0.9792D+00 0.2114D+01

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