Finally consider the state of stresses: FP8 0ox=0 0oy=0 Tox=1 (MPa) Following the same procedure,one obtains andin the orthotropic axes of each ply for a global stress applied on the laminate,and that is reduced to toy =1 MPa.10 It is then easy to determine by simple rule of proportion (or multiplication) the quantities(ob),(ob),and (Te,b)in each ply,corresponding to loadings that are no longer unitary,but equal successively to N:=(Goxb) then: Ny=(ob) then: Tsy =(toxb) Subsequently,the principle of superposition allows one to determine (b)oal(b)a and (Tb)oal when one applies simultaneously NN and To From these it is possible to write the modified Hill-Tsai expression in the form of Equation 12.10,which will provide the thickness for the laminate needed to avoid the fracture of the ply under consideration. If b is the laminate thickness obtained from the ply number k,after having gone over all the plies,one will retain for the final thickness b the thickness of highest value found as: b=sup (by2 Remark:The principle of calculation is conserved when the plies have different thicknesses with any orientations.It then becomes indispensable to program the procedure,or to use existing computer programs.Then one can propose a complete composition for the laminate and verify that the solution is satisfactory regarding the criterion mentioned previously (deformation and fracture).This is This calculation can be easily programmed on a computer:cf.Application 18.2.2Program for Calculation of a Laminate."One will find in Appendix 1 at the end of the book the values o,o Te obtained for the particular case of a carbon/epoxy laminate with ply orientations of0°，90°，+45°，-45°.These values are given in Plates1to12. For example,one has the following: ox=1MPa→oi,oi,tta oax(MPa)→ot,C,Tu then: g-g→=x,and bo=x Gax Ot 12This method to determine the thickness is illustrated by an example:See Application 18.1.6. 2003 by CRC Press LLC
Finally consider the state of stresses: Following the same procedure, one obtains , , and in the orthotropic axes of each ply for a global stress applied on the laminate, and that is reduced to MPa.10 It is then easy to determine by simple rule of proportion (or multiplication)11 the quantities (s
h), (st h), and (t
th) in each ply, corresponding to loadings that are no longer unitary, but equal successively to then: then: Subsequently, the principle of superposition allows one to determine (s
h)total (st h)total and (t
t h)total when one applies simultaneously Nx, Ny, and Txy. From these it is possible to write the modified Hill–Tsai expression in the form of Equation 12.10, which will provide the thickness for the laminate needed to avoid the fracture of the ply under consideration. If hk is the laminate thickness obtained from the ply number k, after having gone over all the plies, one will retain for the final thickness h the thickness of highest value found as: h = sup {hk} 12 Remark: The principle of calculation is conserved when the plies have different thicknesses with any orientations. It then becomes indispensable to program the procedure, or to use existing computer programs. Then one can propose a complete composition for the laminate and verify that the solution is satisfactory regarding the criterion mentioned previously (deformation and fracture). This is 10 This calculation can be easily programmed on a computer: cf. Application 18.2.2 “Program for Calculation of a Laminate.” One will find in Appendix 1 at the end of the book the values s
, st , t
t obtained for the particular case of a carbon/epoxy laminate with ply orientations of 0∞, 90∞, +45∞, -45∞. These values are given in Plates 1 to 12. 11 For example, one has the following: 12 This method to determine the thickness is illustrated by an example: See Application 18.1.6. sox ¢¢¢ = 0 soy ¢¢¢ = 0 t oxy ¢¢¢ = 1 MPa ( ) s
¢¢ st ¢¢ t
t ¢¢ t oxy ¢¢ = 1 s ox¢ 1 MPa s
¢, s t ¢, t
t = Æ ¢ sox( )Æ MPa s
, st, t
t then: sox s ox¢ -------- s
s
¢ ----- fi s
s
¢ sox 1 ¥ -------, and hs
s
¢ 1 == = ------- ¥ Nx Nx = ( ) soxh Ny = ( ) soyh Txy = ( ) t oxyh TX846_Frame_C12 Page 243 Monday, November 18, 2002 12:27 PM © 2003 by CRC Press LLC