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Chapter 8.  Optimal composite structures       369

          i.e., each layer with angle +4i should be accompanied with the layer of the same
          thickness but with angle -4i.
            Consider,  for  example,  the  uniform  tension  such  that  iVr  = N,. = N, N,.  = 0,
          n, = 1, n,.   = 0, A = 0.5. For this case, Eqs. (8.12) and (8.13)  yield


                                                                            (8.15)


          The natural structure for this case corresponds to the cross-ply laminate for which
          k = 2,  4, = 0", 42 = 90"  (Fig. 8.2(a)). Then,  the  second  equation  of  Eqs. (8.15)
          gives the evident result hl  = 62.
            Consider the first equation from which it follows that the total  thickness of  the
          optimal laminate is twice as high as the thickness of the metal plate under the same
          loading  conditions.  This  result  is  quite  natural  because, in  contrast  to  isotropic
          materials, the monotropic layer can work  only in one direction - along the fibers.
          So, we need to have the 0"-layer to take N, = N  and the same, but 90"-layer to take
          N,, = N. From this we can conclude that the directional character of a composite ply
          stiffness and strength is actually the material shortcoming rather than its advantage.
          Real  advantages  of  composite  materials  are  associated  with  their  high  specific
          strength provided by thin fibers (see Section 3.2. l), and if we had isotropic materials
          with such specific strength, no composites would be developed and implemented.
            Return  to  the  second  equation  of  Eqs. (8.15)  which  shows  that  in  addition
          to  a  cross-ply  laminate  there  exists  an  infinite  number  of  optimal  structures.
          For  example,  this  equation  is  satisfied  for  a  symmetric  f45"angle-ply  laminate
          (Fig. 8.2b). Moreover, all the quasi-isotropic laminates discussed in Section 5.5 and
          listed in Table 5.1 satisfy the optimality conditions for uniform tension.
            A loading case, important for applications, corresponds to a cylindrical pressure
          vessel considered in Section 6.3. Winding of such a vessel is shown in Fig. 7.43. For
          this type of loading




          where N, and N,, are the circumferential and the axial stress resultants, respectively,
          p  the  internal  pressure  and  R  is  the  cylinder radius.  Thus,  we  have  n,, = 2  and














                Fig. 8.2.  Cross-ply (a) and f45" angle-ply (b) optimal structures for uniform  tension.
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