\(\( (*\ \ DADOS\ *) 

     (*\ NN\  = \ 4\ ; 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ Numero\ de\ laminas\ *) 
teta = {0, 90, 90, 0} Pi/180; 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ Angulos\ das\ laminas.\ A\ lamina\ 1\ e' inferior, \ 
        com\ cota\ z\ \(negativa.\)\ *) 

      iso = {"orto", "orto", "orto", "orto"}; \ \ 
       (*\ indicar\ se\ cada\ lamina\ e'\ ortotropica\ ou\ isotropica\ 

        	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        usando\ orto\ ou\ iso\ *) 
e1 = {1, 1, 1, 1} 138.0\ 10\^3; 
       (*\ Modulo\ na\ direcao\ 1\ de\ cada\ lamina.\ MPa\ *) 

      e2 = {\ 1, 1, 1, 1} 9300; \ \ \ \ \ \ \ \ 
       (*\ Modulo\ na\ direcao\ 2\ de\ cada\ lamina\ *) 

      g12 = {1, 1, 1, 1} 4600; \ \ \ \ \ \ 
       (*\ Modulo\ na\ direcao\ 12\ de\ cada\ lamina\ *) 

      nu13 = {1, 1, 1, 1} 0.3; 
nu23 = {1, 1, 1, 1} 0.5; 

      nu12 = {1, 1, 1, 1} 0.3; 
\ \ \ \ \ \ 
       (*\ Coef.Poisson\ na\ direcao\ 12\ de\ cada\ lamina\ *) 

      h = {1, 1, 1, 1} 0.25; 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ Espessuras\ das\ laminas.\ mm.\ *) 
g13 = {1, 1, 1, 1} 4600; 
      \ \ \ \ \ \ \  (*\ Modulos\ \(transversais.\)\ \ \ \ \ \ \ \ *) 

      g23 = \ {1, 1, 1, 1} 3100; 
e3 = e2; 
       (*\ Coef.Poisson\ na\ direcao\ 23\ de\ cada\ lamina\ *) 
mh\  = \ 1; 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ numero\ de\ termos\ na\ serie\ em\ x\ *) 
nh\  = \ 1; 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ numero\ de\ termos\ na\ serie\ em\ y\ *) 
q0\  = \ \(-1\); 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ Carga\ distribuida\ uniforme, \ F/Area\ *) 
a = 4; 
b = 4; 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ Lados\ da\ placa.\ a\ na\ direcao\ x\ *) 
xc = \ \ 1.001\ a/2; 
      \ \ \ \ \  (*\ Ponto\ para\ o\ calculo\ das\ tensoes\ coplanares*) 

      yc = \ \ 0.001\ b/2; 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *)  (*\ 
      A\ matriz\ torna - se\ singular\ em\ xc = 
        \(a/2\ ou\ yc = b/2.\ Para\ obter\ respostas\ nestes\)

      	pontos\ usar\ por\ exemplo\ xc = \ \ 1.001\ a/2.\ \ *) 

     (*\ \( ===  ===  ===  ===  ===  ===  ===  ===  ===  ===  ===  ===  === 
           ===  ===  ===  ===  ===  == 
		 ===  ===  ===  ===  ===  === 
           ===  ===  ===  ===  ===  ===  ===  ===  ===  ===  ===  ===  === 
          \) = \ \ *) \ \ \ \)\)

\(\( (*\ \ DADOS\ de\ reserva\ \ \ \ *) 
\ NN\  = \ 4\ ; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ Numero\ de\ laminas\ *) 

    teta = {0, 90, 90, 0} Pi/180; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Angulos\ das\ laminas.\ A\ lamina\ 1\ e' inferior, \ 
      com\ cota\ z\ \(negativa.\)\ *) 

    iso = {"orto", "orto", "orto", "orto"}; \ \ 
     (*\ indicar\ se\ cada\ lamina\ e'\ ortotropica\ ou\ isotropica\ 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
      usando\ orto\ ou\ iso\ *) 
e2 = {\ 6890, 6890, 6890, \ 6890}; 
    \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Modulo\ na\ direcao\ 2\ de\ cada\ lamina\ *) 
e1 = 25\ e2; 
    \ \ \ \ \ \ \  (*\ Modulo\ na\ direcao\ 1\ de\ cada\ lamina.\ MPa\ *) 

    g12 = 0.5\ e2; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Modulo\ na\ direcao\ 12\ de\ cada\ lamina\ *) 

    nu12 = {0.25, 0.25, \ 0.25, \ 0.25}; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Coef.Poisson\ na\ direcao\ 12\ de\ cada\ lamina\ *) 

    h = {10, 10, 10, 10\ }/4; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ Espessuras\ das\ laminas.\ mm.\ *) 
g13 = 0.5\ e2; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Modulos\ \(transversais.\)\ \ \ \ \ \ \ \ *) 
g23 = \ 0.2\ e2; 

    e3 = {\ 6890, 6890, 6890, \ 6890}; 
nu13 = {0.25, \ 0.25, 0.25, \ 0.25}; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Coef.Poisson\ na\ direcao\ 13\ de\ cada\ lamina\ *) 

    nu23 = {0.25, \ 0.25, 0.25, \ 0.25}; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Coef.Poisson\ na\ direcao\ 23\ de\ cada\ lamina\ *) 
mh\  = \ 1; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ numero\ de\ termos\ na\ serie\ em\ x\ *) 
nh\  = \ 1; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ numero\ de\ termos\ na\ serie\ em\ y\ *) 
q0\  = \ 0.001; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Carga\ distribuida\ uniforme, \ F/Area\ *) 
a = 1000; 
b = 1000; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Lados\ da\ placa.\ a\ na\ direcao\ x\ *) 
xc = \ \ 1.001\ a/2; 
    \ \ \ \ \  (*\ Ponto\ para\ o\ calculo\ das\ tensoes\ coplanares*) 

    yc = \ \ 0.001\ b/2; \ \ \ \ \ \ 
     (*\ A\ matriz\ torna - se\ singular\ em\ xc = 
        \(a/2\ ou\ yc = b/2.\ Para\ obter\ respostas\ nestes\)

      	pontos\ usar\ por\ exemplo\ xc = \ \ 1.001\ a/2.\ \ *) \ \ \ \ \ 
    \)\)

\(\( (*\ CALCULOS\ 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ COTAS\ *) 
z = Table[0, {j, NN + 1}]; 

    z0 = Table[0, {j, NN}]; 
\ \ \ \ \ \ \ \ \ 
     (*\ Cotas\ da\ superficie\ inferior\ de\ cada\ lamina\ e\ de\ sua\ 
        sup.media\ *) 
H = Sum[h[\([j]\)], {j, 1, NN}]; 
    \ \ \ \ \ \ \ \ \ \ \ \  (*\ espessura\ total\ do\ laminado\ *) 

    z[\([1]\)] = \(-H\)/2; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ cota\ sup.inf.\ do\ laminado\ *) 

    Do[z[\([j + 1]\)] = z[\([j]\)] + h[\([j]\)]; 
	\ \ \ 
      z0[\([j]\)] = z[\([j]\)] + h[\([j]\)]/2, {j, 1, NN}]; \)\)

H

\(\( (*\ Inicializar\ areas\ *) 
c1 = Table[0, {i, 9}, {k, NN}]; 
    \ \ \ \ \  (*\ Matriz\ C\ de\ rigidez\ de\ cada\ lamina, \ 
	
      3  D.\ Valores\ em\ coluna\ da\ parte\ triang.sup.\( : \ c11\), c12, 
      c22, c13, c23, c33, \ C44, CC55, c66\ \ *) 

    s1 = Table[0, {i, 3}, {j, 3}]; \ \ \ \ \ \ \ 
     (*\ area\ de\ trabalho\ para\ flexibilidade\ *) 

    c1t = Table[0, {i, 3}, {j, 3}]; \ \ \ \ \ \ \ 
     (*\ area\ de\ trabalho\ para\ rigidez\ *) 

    cx = Table[0, {i, 3}, {j, 3}]; \ \ \ \ \ \ \ 
     (*\ area\ de\ trabalho\ para\ rigidez\ nas\ direcoes\ x - y\ *) \)\)

\(\(\(Do[
      s1 = {{1\/e1[\([j]\)], \(-nu12[\([j]\)]\)\/e1[\([j]\)], 
            \(-nu13[\([j]\)]\)\/e1[\([j]\)]}, 

          			\ \ \ \ \ \ \ \ \ \ \ \ {\(-nu12[\([j]\)]\)\/e1[\([j]\)], 
            1\/e2[\([j]\)], \(-nu23[\([j]\)]\)\/e2[\([j]\)]}, 

          \ \ \ \ \ \ \ \ \ \ 			\ {\(-nu13[\([j]\)]\)\/e1[\([j]\)], 
            \(-nu23[\([j]\)]\)\/e2[\([j]\)], 1\/e3[\([j]\)]}}; 
	\ \ \ \ 
      c1t = Inverse[s1]; 
	\ \ \ \  (*\ rotacao\ para\ matriz\ cx\ \ *) 

      \ \ \ \ \ \ \ c4 = Cos[teta[\([j]\)]\ ]^4; \ \ \ \ \ \ \ \ \ 
		\ \ 
      s4 = Sin[teta[\([j]\)]\ ]^4; 
	\ \ \ \ c2 = Cos[teta[\([j]\)]\ ]^2; 
      \ \ \ \ \ \ \ \ \ \ 
	\ 	\ s2 = Sin[teta[\([j]\)]\ ]^2; 
	
      c1[\([1, j]\)] = 
        c1t[\([1, 1]\)] c4 + 
          2 \((c1t[\([1, 2]\)] + 2  g12[\([j]\)])\) s2\ c2 + 
          c1t[\([2, 2]\)] s4; 
	
      c1[\([3, j]\)] = 
        c1t[\([1, 1]\)] s4 + 
          2 \((c1t[\([1, 2]\)] + 2  g12[\([j]\)])\) s2\ c2 + 
          c1t[\([2, 2]\)] c4; 
	
      c1[\([9, j]\)] = 
        \((c1t[\([1, 1]\)] + c1t[\([2, 2]\)] - 2  c1t[\([1, 2]\)] - 
                2\ g12[\([j]\)])\) s2\ c2 + g12[\([j]\)] \((s4 + c4)\); 
	
      c1[\([2, j]\)] = 
        \((c1t[\([1, 1]\)] + c1t[\([2, 2]\)] - 4\ g12[\([j]\)])\) s2\ c2 + 
          c1t[\([1, 2]\)] \((s4 + c4)\); 
	
      c1[\([4, j]\)] = c1t[\([1, 3]\)] c2 + c1t[\([2, 3]\)] s2; 
	
      c1[\([5, j]\)] = c1t[\([1, 3]\)] s2 + c1t[\([2, 3]\)] c2; 
	
      c1[\([6, j]\)] = c1t[\([3, 3]\)]; 
	
      c1[\([7, j]\)] = g23[\([j]\)] c2 + g13[\([j]\)] s2; 
	
      c1[\([8, j]\)] = g23[\([j]\)] s2 + g13[\([j]\)] c2, 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {j, NN}]; \)\ \)\)

TableForm[s1]TableForm[c1t]TableForm[c1]

\(\( (*\ HARMONICOS\ DO\ CARREGAMENTO\ *) 

    qmn = Table[0, {i, mh}, {j, nh}]; 
 (*\ CARGA\ UNIFORME\ q0\ *) 

     (*\ \ Do[
\ \ \ \ 	
        Do[qmn[\([i, j]\)] = \(16\ q0\)\/\(Pi\^2\ i\ j\), {i, 1, mh, 2}], {j, 
          1, nh, 2}]; \ \ \ \ \ \ \ *) 

     (*\ CARGA\ SENOIDAL\ q0\ \ USADO\ SO\ NO\ CASO\ mh\  = \ 
        \(nh\  = \ 1\)\ \ \ \ \ *) 
\ 
    Do[
\ \ \ \ 	Do[qmn[\([i, j]\)] = q0, {i, 1, mh, 2}], {j, 1, nh, 2}]; 
    \ \ 
TableForm[qmn]\)\)

\(\( (*\ 
      INICIO\ DA\ 
        \(\(\(\(\(\(\(\(\(\(\(\(\(\(SOLUCAO\ --\)--\)--\)--\)--\)--\)--\)--
                      \)--\)--\)--\)--\)--\)--\)*) 

    ak = Table[0, {i, mh}, {j, nh}, {k, NN}]; 

    bk = Table[0, {i, mh}, {j, nh}, {k, NN}]; 

    ck = Table[0, {i, mh}, {j, nh}, {k, NN}]; 

    dk = Table[0, {i, mh}, {j, nh}, {k, NN}]; 

    gk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    mk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    al = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    rk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    lk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    mij = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}, {jj, 3}]; 

    chi = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    chs = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    shi = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    shs = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 3}]; 

    jk = Table[0, {i, 3}]; 

     (*\ Areas\ para\ laminas\ isotropicas.\ Para\ cada\ lamina\ k, \ 
      ha\ duas

      	\ \ linhas\ de\ seis\ termos, \ 
      a\ linha\ 1\ com\ valores\ na\ sup.inf.e\ 

      	\ \ linha\ 2\ na\ superficie\ superior.\ uk, \ vk, \ 
      wk\ definem\ deslocamentos

      	\ \ e\ ul, \ vl, \ 
      wl\ definem\ as\ derivadas\ em\ z.\ Sao\ funcoes\ apenas\ de\ z\ *) 

    uk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 2}, {jj, 6}]; 

    vk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 2}, {jj, 6}]; 

    wk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 2}, {jj, 6}]; 

    uk = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 2}, {jj, 6}]; 

    ul = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 2}, {jj, 6}]; 

    vl = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 2}, {jj, 6}]; 

    wl = Table[0, {i, mh}, {j, nh}, {k, NN}, {ii, 2}, {jj, 6}]; \)\)

\(\(Do\ [
\ \ \ \ \ 
      Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ percorre\ cada\ harmonico\ m - n\ \ *) 
\ \ \ \ \ \ \ 
        pm = \(n\ Pi\)\/a; \ \ \ \ \ pn = \(m\ Pi\)\/b; 
\ \ \ \ \ \ 
        Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ percorre\ as\ laminas\ *) 
	\ \ \ \ \ \ \ \ 
          ak[\([m, n, k]\)] = c1[\([6, k]\)]\ c1[\([7, k]\)]\ c1[\([8, k]\)]; 
          
		\ \ \ \ \ 
          bk[\([m, n, k]\)] = 
            \(pm\^2\) 
                \((c1[\([7, k]\)] 
                      \((c1[\([1, k]\)] c1[\([6, k]\)] - c1[\([4, k]\)]\^2)
                        \) + c1[\([8, k]\)] 
                      \((c1[\([6, k]\)] c1[\([9, k]\)] - 
                          2  c1[\([4, k]\)] c1[\([7, k]\)])\)\ )\) + 

              				\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(pn\^2\) \((c1[\([8, k]\)] 
                      \((c1[\([3, k]\)] c1[\([6, k]\)] - c1[\([5, k]\)]\^2)
                        \) + c1[\([7, k]\)] 
                      \((c1[\([6, k]\)] c1[\([9, k]\)] - 
                          2  c1[\([5, k]\)] c1[\([8, k]\)])\))\); 

          		\ \ \ \ 
          ck[\([m, n, k]\)] = 
            \(-pm\^4\) 
                \((c1[\([9, k]\)] 
                      \((c1[\([1, k]\)] c1[\([6, k]\)] - c1[\([4, k]\)]\^2)
                        \) + c1[\([8, k]\)] 
                      \((c1[\([1, k]\)] c1[\([7, k]\)] - 
                          2  c1[\([4, k]\)] c1[\([9, k]\)])\))\) + 

              				\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
              \(pm\^2\) \(pn\^2\) 
                \((\(-c1[\([1, k]\)]\) 
                      \((c1[\([3, k]\)] c1[\([6, k]\)] - c1[\([5, k]\)]\^2)
                        \) - \ 2 
                      \((c1[\([2, k]\)] + 

                          						\ \ \ \ \ \ \ \ \ 	\ \ \ \ \ \ \ \ 
                          c1[\([9, k]\)])\) 
                      \((c1[\([4, k]\)] + c1[\([8, k]\)])\) 
                      \((c1[\([5, k]\)] + c1[\([7, k]\)])\) - 

                    						\ \ \ \ \ \ \ \ \ 	\ \ \ \ \ \ \ \ 
                    2  c1[\([7, k]\)] c1[\([8, k]\)] c1[\([9, k]\)] + 
                    2  c1[\([1, k]\)] c1[\([5, k]\)] c1[\([7, k]\)] + 

                    						\ \ \ \ \ \ \ \ \ 	\ \ \ \ \ \ \ \ 
                    c1[\([2, k]\)] c1[\([6, k]\)] 
                      \((c1[\([2, k]\)] + 2  c1[\([9, k]\)])\) + 

                    					\ \ \ \ \ \ 	\ \ \ \ 	\ \ \ \ \ \ \ \ 
                    c1[\([4, k]\)] c1[\([3, k]\)] 
                      \((c1[\([4, k]\)] + 2  c1[\([8, k]\)])\))\) - 

              			\ \ \ \ \ \ \ \ \ \ \ 	\ \ \ \ \ 
              \(pn\^4\) 
                \((c1[\([9, k]\)] 
                      \((c1[\([3, k]\)] c1[\([6, k]\)] - c1[\([5, k]\)]\^2)
                        \) + 

                    						\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 	\ \ \ \ \ \ c1[\([7, k]\)] \((c1[\([3, k]\)] c1[\([8, k]\)] - 
                          2  c1[\([5, k]\)] c1[\([9, k]\)])\))\); 
		\ \ \ 
          dk[\([m, n, k]\)] = 
            \(pm\^6\) c1[\([1, k]\)] c1[\([8, k]\)] c1[\([9, k]\)] + 
              \(pm\^4\) \(pn\^2\) 
                \((c1[\([8, k]\)] 
                      \((c1[\([1, k]\)] c1[\([3, k]\)] - c1[\([2, k]\)]\^2)
                        \) + 
						\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 	
                    \(+c1[\([9, k]\)]\) 
                      \((c1[\([1, k]\)] c1[\([7, k]\)] - 
                          2  c1[\([2, k]\)] c1[\([8, k]\)])\))\) + 

              				\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
              \(pm\^2\) \(pn\^4\) 
                \((c1[\([7, k]\)] 
                      \((c1[\([1, k]\)] c1[\([3, k]\)] - c1[\([2, k]\)]\^2)
                        \) + 
						\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
                    c1[\([9, k]\)] 
                      \((c1[\([3, k]\)] c1[\([8, k]\)] - 
                          2  c1[\([2, k]\)] c1[\([7, k]\)])\))\) + 

              				\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
              \(pn\^6\) c1[\([3, k]\)] c1[\([7, k]\)] c1[\([9, k]\)]; 
			
          
			ddk = 
            \(-\(\(3  ck[\([m, n, k]\)]\ ak[\([m, n, k]\)] + 
                    bk[\([m, n, k]\)]\ bk[\([m, n, k]\)]\)\/\(3 
                     ak[\([m, n, k]\)]\^2\)\)\); \ \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ Constantes\ da\ forma\ padrao\ da\ eq.caracteristica\ *) 

          			fk = 
            \(-\(1\/\(27  ak[\([m, n, k]\)]\^3\)\)\) 
              \((2\ bk[\([m, n, k]\)]\^3 + 
                  9  ak[\([m, n, k]\)]\ bk[\([m, n, k]\)]\ ck[\([m, n, k]\)]\ 
                   + \ 27  dk[\([m, n, k]\)]\ ak[\([m, n, k]\)]\^2)\); 

          			hk = fk\^2\/4 + ddk\^3\/27; \ 
          Print["fk2 ddk3 hk ", fk\^2\/4, ddk\^3\/27, hk]; 
			

          			 (*\ TESTE\ SE\ LAMINA\ E'\ ORTOTROPICA\ OU\ ISOTROPICA\ *) 

          		If[iso[\([k]\)] == "orto", 
			
            ang = \(\(-\@27\) fk\)\/\(2 \((\(-ddk\))\)\^\(3/2\)\); 
			
             (*\ If[Abs[hk] > 1/Abs[ak[\([m, n, k]\)]], fi = ArcCos[ang], 
                fi = Pi/1000]; \ *) 
\ \ \ \ \ \ \ \ fi = ArcCos[ang]; 

            					\ \ Print["ang ", ang, " fi ", fi]; 
			

            			Do[
			
              gk[\([m, n, k, j]\)] = 
                2 \(\@\(\(-ddk\)/3\)\) Cos[\((fi + 2 \((j - 1)\) Pi)\)/3]; 
              \ \ \ \ \ \ \ \ \ \  (*\ Autovalores\ *) 
			
              mk[\([m, n, k, j]\)] = 
                \@Abs[gk[\([m, n, k, j]\)] + 
                      bk[\([m, n, k]\)]\/\(3\ ak[\([m, n, k]\)]\)]; 	
			
              If[gk[\([m, n, k, j]\)] + 
                    bk[\([m, n, k]\)]\/\(3\ ak[\([m, n, k]\)]\) > 0, 

                				\ \ \ \ \ \ \ \ \ \ al[\([m, n, k, j]\)] = 1; 

                				\ \ \ \ \ \ \ \ \ 
                chi[\([m, n, k, j]\)] = 
                  Cosh[mk[\([m, n, k, j]\)] z[\([k]\)]\ ]; \ \ \ \ \ \ 
                 (*\ superficie\ inf\ da\ lam\ k\ *) 

                				\ \ \ \ \ \ \ \ \ 
                chs[\([m, n, k, j]\)] = 
                  Cosh[mk[\([m, n, k, j]\)] z[\([k + 1]\)]\ ]; \ 
                 (*\ superficie\ sup\ da\ lam\ k\ *) 

                			\ \ \ \ \ \ \ \ \ 	
                shi[\([m, n, k, j]\)] = 
                  Sinh[mk[\([m, n, k, j]\)] z[\([k]\)]\ ]; 

                				\ \ \ \ \ \ \ \ \ 
                shs[\([m, n, k, j]\)] = 
                  Sinh[mk[\([m, n, k, j]\)] z[\([k + 1]\)]\ ], 
				

                				\ \ \ \ \ \ \ \ \ \ al[\([m, n, k, j]\)] = \(-1\); 

                				\ \ \ \ \ \ \ \ \ 
                chi[\([m, n, k, j]\)] = 
                  Cos[mk[\([m, n, k, j]\)] z[\([k]\)]\ ]; 

                				\ \ \ \ \ \ \ \ \ 
                chs[\([m, n, k, j]\)] = 
                  Cos[mk[\([m, n, k, j]\)] z[\([k + 1]\)]\ ]; 

                			\ \ \ \ \ \ \ \ \ 	
                shi[\([m, n, k, j]\)] = 
                  Sin[mk[\([m, n, k, j]\)] z[\([k]\)]\ ]; 

                				\ \ \ \ \ \ \ \ \ 
                shs[\([m, n, k, j]\)] = 
                  Sin[mk[\([m, n, k, j]\)] z[\([k + 1]\)]\ ]]; 
			

              \ \ \ \ 	
              jk[\([j]\)] = 
                c1[\([6, k]\)] c1[\([7, k]\)] mk[\([m, n, k, j]\)]\^4 + 
                  al[\([m, n, k, j]\)]\ \(mk[\([m, n, k, j]\)]\^2\) 
                    \((\(pn\^2\) 
                          \((c1[\([5, k]\)]\^2 - 
                              c1[\([3, k]\)] c1[\([6, k]\)] + 

                              			\ \ \ \ \ \ \ \ \ 
                              2  c1[\([5, k]\)] c1[\([7, k]\)])\) - 
                        \(pm\^2\) 
                          \((c1[\([7, k]\)] c1[\([8, k]\)] + 
                              c1[\([6, k]\)] c1[\([9, k]\)])\))\) + 

                  	\ \ \ \ \ \ \ \ \ \ \ \ \ 
                  \((c1[\([9, k]\)]\ pm\^2 + c1[\([3, k]\)] pn\^2)\) 
                    \((c1[\([8, k]\)] pm\^2 + c1[\([7, k]\)] pn\^2)\); 

              \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ 
              rk[\([m, n, k, j]\)] = 
                \(\(pm\ mk[\([m, n, k, j]\)]\)\/jk[\([j]\)]\) 
                  \((al[\([m, n, k, j]\)]\ \ mk[\([m, n, k, j]\)]\^2\ 
                        c1[\([7, k]\)] 
                        \((c1[\([4, k]\)] + c1[\([8, k]\)])\) - 

                      							
                      \((c1[\([4, k]\)] + c1[\([8, k]\)])\) 
                        \((c1[\([9, k]\)] pm^2 + c1[\([3, k]\)] pn\^2)\) + 

                      			\ \ \ \ \ \ \ \ \ \ 
                      \(pn\^2\) \((c1[\([5, k]\)] + c1[\([7, k]\)])\) 
                        \((c1[\([2, k]\)] + c1[\([9, k]\)])\))\); 
			

              			\ lk[\([m, n, k, j]\)] = 
                \(\(pm\ pn\)\/jk[\([j]\)]\) 
                  \((al[\([m, n, k, j]\)] mk[\([m, n, k, j]\)]\^2\ \ 
                        \((c1[\([6, k]\)] 
                              \((c1[\([2, k]\)] + c1[\([9, k]\)])\) - 
                            \((c1[\([5, k]\)] + 
										
                                  c1[\([7, k]\)])\) 
                              \((c1[\([4, k]\)] + c1[\([8, k]\)])\))\) - 
                      \((c1[\([2, k]\)] + c1[\([9, k]\)])\) 
                        \((c1[\([8, k]\)]\ pm\^2 + 
									\ \ 
                            c1[\([7, k]\)] pn\^2)\))\), 
				{j, 3}]; 

            
\ \ \ \ \ \ \ \ \ \ \ 
            Do[
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
              Do[\ \ mij[\([m, n, k, ii, jj]\)] = 
                  al[\([m, n, k, jj]\)] mk[\([m, n, k, jj]\)] 
                      rk[\([m, n, k, jj]\)] c1[\([3 + ii, k]\)] - 

                    							\ \ \ \ \ \ \ \ \ \ 
                    pm\ c1[\([1 + ii \((ii - 1)\)/2, k]\)] - 
                    pn\ c1[\([2 + ii \((ii - 1)\)/2, k]\)]\ 
                      lk[\([m, n, k, jj]\)], 

                \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {ii, 3}], {jj, 3}]

            			\ \ \ \ \ \ \ \ \ \ \ ]; 
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ fim\ do\ if\ se\ a\ lamina\ e'\ ortotropica.\ Agora\ ver\ se\ e' iso
              \ *) 
		
		
          If[iso[\([k]\)] == "iso", 		
				Print["x "]; 

            			\ Do[c = \@\(pm\^2 + pn\^2\); 
					
              chi[\([m, n, k, j]\)] = Exp[c\ z[\([k]\)]\ \ \ ]; \ \ \ \ \ \ 
               (*\ superficie\ inf\ da\ lam\ k\ *) 
				\ \ \ 
              chs[\([m, n, k, j]\)] = Exp[c\ z[\([k + 1]\)]\ ]; \ 
               (*\ superficie\ sup\ da\ lam\ k\ *) 
				\ \ 
              shi[\([m, n, k, j]\)] = 1/Exp[c\ z[\([k]\)]\ \ ]; \ \ \ \ \ 

              				\ \ \ 
              shs[\([m, n, k, j]\)] = 1/Exp[c\ z[\([k + 1]\)]\ \ ]\ , 
              \ 	{j, 3}]; 
			
					
             (*\ deslocamentos\ e\ derivadas, \ parte\ dependente\ de\ z\ *) 
              
				
            ciso\  = 
              \((\(c1[\([2, k]\)] - 3  c1[\([1, k]\)]\)\/\(c1[\([2, k]\)] + 
                    c1[\([1, k]\)]\))\); 
					
            ci = Exp[c\ z[\([k]\)]\ \ \ ]; 
				
            si = 1/Exp[c\ z[\([k]\)]\ \ \ ]; 
\ \ \ \ \ \ \ \ \ \ 
            cs = Exp[c\ z[\([k + 1]\)]\ ]; 
				
            ss = 1/Exp[c\ z[\([k + 1]\)]\ ]; \ \ \ \ 
			
				\ 
            uk[\([m, n, k]\)]
                 = {{ci, si, z[\([k]\)] ci, z[\([k]\)] si, 0, 0}, 

                							{cs, ss, z[\([k + 1]\)] cs, z[\([k + 1]\)] ss, 
                  0, 0}}; 
			\ 
            vk[\([m, n, k]\)]
                 = {{0, 0, \(pn\/pm\) z[\([k]\)] ci, 
                  \(pn\/pm\) z[\([k]\)] si, si, ci}, 

                							{0, 0, \(pn\/pm\) z[\([k + 1]\)] cs, 
                  \(pn\/pm\) z[\([k + 1]\)] ss, ss, cs}}; 
	
            wk[\([m, n, k]\)]
                 = {{\(pm\/c\) ci, \(\(-pm\)\/c\) si, 
                  \(\((ciso + c\ z[\([k]\)])\)\/pm\) ci, 
                  \(\((ciso - c\ z[\([k]\)])\)\/pm\) si, \(\(-pn\)\/c\) si, 
                  \(pn\/c\) ci}, 

                			{\(pm\/c\) cs, \(\(-pm\)\/c\) ss, 
                  \(\((ciso + c\ z[\([k + 1]\)])\)\/pm\) cs, 
                  \(\((ciso - c\ z[\([k + 1]\)])\)\/pm\) ss, 
                  \(\(-pn\)\/c\) ss, \(pn\/c\) cs}}; 
						
            ul[\([m, n, k]\)] = {
                \ {c\ ci, \(-c\)\ si, \((1 + c\ z[\([k]\)])\) ci, 
                  \((1 - c\ z[\([k]\)])\) si, 0, 0}, 

                							{c\ cs, \(-c\)\ shs[\([m, n, k, 1]\)], 
                  \((1 + c\ z[\([k + 1]\)])\) cs, 
                  \((1 - c\ z[\([k + 1]\)])\) ss, 0, 0}}; 
						
            vl[\([m, n, k]\)]
                 = {{0, 0, \(pn\/pm\) \((1 + c\ z[\([k]\)])\) ci, 
                  \(pn\/pm\) \((1 - c\ z[\([k]\)])\) si, \(-c\)\ si, c\ ci}, 
                
							{0, 0, 
                  \(pn\/pm\) \((1 + c\ z[\([k + 1]\)])\) cs, 
                  \(pn\/pm\) \((1 - c\ z[\([k + 1]\)])\) ss, \(-c\)\ ss, 
                  c\ cs}}; 
	
            wl[\([m, n, k]\)]
                 = {{pm\ ci, pm\ si, 
                  \(\(c \((1 + \ ciso + c\ z[\([k]\)])\)\)\/pm\) ci, 
                  \(\(\(-c\) \((1 + ciso - c\ z[\([k]\)])\)\)\/pm\) si, 
                  pn\ si, pn\ ci}, 

                		{pm\ cs, pm\ ss, 
                  \(\(c \((1 + \ ciso + c\ z[\([k + 1]\)])\)\)\/pm\) cs, 
                  \(\(\(-c\) \((1 + ciso - c\ z[\([k + 1]\)])\)\)\/pm\) ss, 
                  pn\ ss, pn\ cs}}; 
						\ 
            Print[" raiz c de lamina iso ", c]\ \ ]\ , \ \ \ 
				
           (*\ fim\ do\ if\ se\ a\ lamina\ e' \(isotropica.\)\ *) \ \ 


          \ \ \ \ \ \ \ \ \ 
			{k, NN}], {m, mh}], {n, nh}]; \)\)

TableForm[ak]TableForm[bk]TableForm[ck]TableForm[dk]

fkddkfiTableForm[gk]TableForm[mij]

TableForm[mk]\(	TableForm[chs]\)TableForm[shs]

TableForm[uk // N]uk[[1, 1, 2, 2, 4]] // N\(NN
 (*\ TableForm[vk]

      TableForm[wk]\ \ *) \)

\(\( (*\ MATRIZ\ PARA\ DETERMINAR\ OS\ COEFICIENTES\ F1, F2, F3\ E\ G1, 
      G2, G3\ 

      	\ \ \ PARA\ CADA\ LAMINA\ EM\ CADA\ HARMONICO, \ 
      no\ caso\ de\ lamina\ ortotropica, 
	\ \ \ 
      ou\ a11\ a21\ a31\ a41\ a22\ a12\ no\ caso\ de\ lamina\ isotropica, \ 
      nesta\ \(ordem.\)\ *) 
ma = Table[0, {i, 6  NN}, {j, 6  NN}]; \ 
     (*\ matriz\ de\ rigidez\ *) 
vf = Table[0, {i, 6  NN}]; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ vetor\ forca\ *) 

    vs = Table[0, {i, 6  NN}]; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ vetor\ solucao\ com\ F' s\ e\ G' s\ *) 

    fg = Table[0, {m, mh}, {n, nh}, {k, NN}, {i, 6}]; 
     (*\ armazena\ vetores\ \(sol.\)*) 
sigx = Table[0, {l, 2\ NN}, {j, 2}]; 
     (*\ pares\ z\ sigx\ em\ cada\ interface\ *) 

    sigy = Table[0, {l, 2\ NN}, {j, 2}]; 

    sigz = Table[0, {l, \ NN + 1}, {j, 2}]; 

    tauxy = Table[0, {l, 2\ NN}, {j, 2}]; 

    tauxz = Table[0, {l, \ NN + 1}, {j, 2}]; 

    tauyz = Table[0, {l, \ NN + 1}, {j, 2}]; 

    u = Table[0, {l, \ NN + 1}, {j, 2}]; 

    v = Table[0, {l, \ NN + 1}, {j, 2}]; 

    w = Table[0, {l, \ NN + 1}, {j, 2}]; 
 (*\ valores\ normalizados\ *) 

    sigxN = Table[0, {l, 2\ NN}, {j, 2}]; 
     (*\ pares\ z\ sigx\ em\ cada\ interface\ *) 

    sigyN = Table[0, {l, 2\ NN}, {j, 2}]; 

    sigzN = Table[0, {l, \ NN + 1}, {j, 2}]; 

    tauxyN = Table[0, {l, 2\ NN}, {j, 2}]; 

    tauxzN = Table[0, {l, \ NN + 1}, {j, 2}]; 

    tauyzN = Table[0, {l, \ NN + 1}, {j, 2}]; 

    uN = Table[0, {l, \ NN + 1}, {j, 2}]; 

    vN = Table[0, {l, \ NN + 1}, {j, 2}]; 

    wN = Table[0, {l, \ NN + 1}, {j, 2}]; 
\)\)

\(\(
	
    \(Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
		\ \ 
      Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
         (*\ percorre\ cada\ harmonico\ m - n\ \ *) 
\ \ \ 
        pm = \(n\ Pi\)\/a; \ \ \ \ \ pn = \(m\ Pi\)\/b; 
	\ 
        sm = Sin[pm\ xc]; \ cm = Cos[pm\ xc]; \ sn\  = \ Sin[pn\ yc]; \ 
        cn\  = \ Cos[pn\ yc]; 
\ \ \ 
        Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ percorre\ as\ laminas.\ Interfaces\ SUP.SUPERIOR\ DA\ 
              LAM.k\ *) 
		
          If[iso[\([k]\)] == "orto", 
				kk = \((k - 1)\)*6; 

            \ \ \ \ Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
               (*\ corre\ j = 1, 2, 3\ da\ linha\ da\ matriz\ ma\ *) 

              				\ 
              ma[\([kk + 1, kk + j]\)] = cm\ sn\ chs[\([m, n, k, j]\)]; \ \ \ 
               (*u\ F\ *) 
				\ 
              ma[\([kk + 1, kk + j + 3]\)] = cm\ sn\ shs[\([m, n, k, j]\)]; 
               (*u\ G*) 
			\ 
              ma[\([kk + 2, kk + j]\)] = 
                lk[\([m, n, k, j]\)] sm\ cn\ chs[\([m, n, k, j]\)]; \ \ \ 
               (*v\ F\ *) 
			\ 
              ma[\([kk + 2, kk + j + 3]\)] = 
                lk[\([m, n, k, j]\)] sm\ cn\ shs[\([m, n, k, j]\)]; 
               (*v\ G*) 
			
              ma[\([kk + 3, kk + j]\)] = 
						
                rk[\([m, n, k, j]\)] al[\([m, n, k, j]\)] sm\ sn\ 
                  shs[\([m, n, k, j]\)]; \ \ \  (*w\ F\ *) 
		\ \ 
              ma[\([kk + 3, kk + j + 3]\)] = 
                rk[\([m, n, k, j]\)] sm\ sn\ chs[\([m, n, k, j]\)]; \ \ \ 
               (*w\ G*) 
			
					\ 
               (*\ Tensoes\ sigz, \ tauxz, \ tauyz, \ interfaces, 
                sup.\ superior\ *) 
			
              ma[\([kk + 4, kk + j]\)] = 
                sm\ sn\ mij[\([m, n, k, 3, j]\)] chs[\([m, n, k, j]\)]; 
               (*\ F*) 
			
              ma[\([kk + 4, kk + j + 3]\)] = 
                sm\ sn\ mij[\([m, n, k, 3, j]\)] shs[\([m, n, k, j]\)]; 
               (*\ G*) 
			
              ma[\([kk + 5, kk + j]\)] = 
                cm\ sn\ c1[\([8, k]\)] al[\([m, n, k, j]\)] 
                  \((mk[\([m, n, k, j]\)] + pm\ rk[\([m, n, k, j]\)])\) 
                  shs[\([m, n, k, j]\)]; 
              \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ F*) 
			
              ma[\([kk + 5, kk + j + 3]\)] = 
                cm\ sn\ c1[\([8, k]\)] 
                  \((mk[\([m, n, k, j]\)] + pm\ rk[\([m, n, k, j]\)])\) 
                  chs[\([m, n, k, j]\)]; \ \  (*\ G*) 
			
              ma[\([kk + 6, kk + j]\)] = 
                sm\ cn\ c1[\([7, k]\)] al[\([m, n, k, j]\)] 
                  \((mk[\([m, n, k, j]\)]\ lk[\([m, n, k, j]\)] + 
                      pn\ rk[\([m, n, k, j]\)])\) shs[\([m, n, k, j]\)]; 
               (*\ F*) 
			
              ma[\([kk + 6, kk + j + 3]\)] = 
                sm\ cn\ c1[\([7, k]\)] 
                  \((mk[\([m, n, k, j]\)]\ lk[\([m, n, k, j]\)] + 

                      									pn\ rk[\([m, n, k, j]\)])\) 
                  chs[\([m, n, k, j]\)], 
              \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ G*) 
			
					{j, 3}], Print["a ", NN]

            			]; \ \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ fim\ do\ if\ caso\ lamina\ ortotropica.\ Agora\ lamina\ 
              isotropica\ *) 

\ \ \ \ \ \ \ \ \  (*\ \ iso\ iso\ 1\ \ *) 

          				\ 	\ \ 
          If[\ iso[\([k]\)] == "iso", \ \ \ \ \ \ \ \ \ 
             (*\ iso\ iso\ 1\ *) 
			kk = \((k - 1)\)*6; 
			
            Do[\ ma[\([kk + 1, kk + jj]\)] = 
                cm\ sn\ uk[\([m, n, k, 2, jj]\)]; \ \ \ \ \ \ \ \ \ \ 
               (*u\ \ *) 
		\ \ \ \ 	
              ma[\([kk + 2, kk + jj]\)] = sm\ cn\ vk[\([m, n, k, 2, jj]\)]; 
              \ \ \ \ \ \ \ \ \ \ \  (*v\ \ *) 
			\ \ \ \ 
              ma[\([kk + 3, kk + jj]\)] = sm\ sn\ wk[\([m, n, k, 2, jj]\)]; 
              \ \ \ \ \ \ \ \ \ \  (*w\ \ *) 
			\ \ \ \ 
              ma[\([kk + 4, kk + jj]\)] = 
                sm\ sn\ \((
                    c1[\([1, k]\)] wl[\([m, n, k, 2, jj]\)] - \ 

                      								\ \ \ 
                      c1[\([2, k]\)] 
                        \((pm\ uk[\([m, n, k, 2, jj]\)] + 
                            pn\ vk[\([m, n, k, 2, jj]\)])\))\); \ \ \ \ 
               (*\ sigz\ *) 
			\ \ \ 
              ma[\([kk + 5, kk + jj]\)] = 
                cm\ sn\ \((
                      \((c1[\([1, k]\)] - 
						\ \ \ 	
                            c1[\([2, k]\)])\) 
                        \((ul[\([m, n, k, 2, jj]\)] + 
                            pm\ wk[\([m, n, k, 2, jj]\)])\))\)/2; \ \ \ \ 
               (*\ tauxz\ *) 
			\ \ \ 
              ma[\([kk + 6, kk + jj]\)] = 
                sm\ cn\ \((
                      \((c1[\([1, k]\)] - 
					\ \ \ 		
                            c1[\([2, k]\)])\) 
                        \((vl[\([m, n, k, 2, jj]\)] + 
                            pn\ wk[\([m, n, k, 2, jj]\)])\))\)/2, \ 

              			\ {jj, 6}]\ \ \ \ \ \ \ \ \  (*\ fim\ do\ jj\ \ *) 
, 
            Print["b ", NN]], 
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ fim\ if\ iso\ *) \ \ \ \ \ 
			
	{k, NN - 1}]; 
        \ \ \ \ \ \ \ \ \ \ 
		

        If[iso[\([1]\)] == "orto", 
		
          Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
             (*\ corre\ j = 1, 2, 3\ da\ linha\ da\ matriz\ ma\ *) 
				
             (*\ As\ 3\ linhas\ das\ tensoes\ na\ superficie\ inferior\ do\ 
                laminado\ *) 
			
            ma[\([\((NN - 1)\)*6 + 1, j]\)] = 
              sm\ sn\ mij[\([m, n, 1, 3, j]\)] chi[\([m, n, 1, j]\)]; 
             (*\ F*) 
			
            ma[\([\((NN - 1)\)*6 + 1, j + 3]\)] = 
              sm\ sn\ mij[\([m, n, 1, 3, j]\)] shi[\([m, n, 1, j]\)]; 
             (*\ G*) 
			
            ma[\([\((NN - 1)\)*6 + 2, j]\)] = 
              cm\ sn\ c1[\([8, 1]\)] al[\([m, n, 1, j]\)] 
                \((mk[\([m, n, 1, j]\)] + 
								
                    pm\ rk[\([m, n, 1, j]\)])\) shi[\([m, n, 1, j]\)]; \ 
             (*\ F*) 
			
            ma[\([\((NN - 1)\)*6 + 2, j + 3]\)] = 
              cm\ sn\ c1[\([8, 1]\)] 
                \((mk[\([m, n, 1, j]\)] + 
								
                    pm\ rk[\([m, n, 1, j]\)])\) chi[\([m, n, 1, j]\)]; \ 
             (*\ G*) 
			
            ma[\([\((NN - 1)\)*6 + 3, j]\)] = 
              sm\ cn\ c1[\([7, 1]\)] al[\([m, n, 1, j]\)] 
                \((mk[\([m, n, 1, j]\)]\ lk[\([m, n, 1, j]\)] + 
                    pn\ rk[\([m, n, 1, j]\)])\) shi[\([m, n, 1, j]\)]; \ 
             (*\ F*) 
			
            ma[\([\((NN - 1)\)*6 + 3, j + 3]\)] = 
              sm\ cn\ c1[\([7, 1]\)] 
                \((mk[\([m, n, 1, j]\)]\ lk[\([m, n, 1, j]\)] + 

                    								pn\ rk[\([m, n, 1, j]\)])\) 
                chi[\([m, n, 1, j]\)], \ \ \  (*\ G*) 
					{j, 3}]

          			, Print["c ", NN]]; \ \ \ \ \ \ \ \ \ \ \ \ \ 
         (*\ fim\ do\ if\ caso\ lamina\ \(ortotropica.\)\ \ \ \ \ \ \ \ *) 
          \ \ \ \ \ 
							
		
	
        If[iso[\([NN]\)] == "orto", 
           (*\ As\ 3\ linhas\ das\ tensoes\ na\ superficie\ superior\ do\ 
              \(lam.\)\ *) 
			
          Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
             (*\ corre\ j = 1, 2, 3\ da\ linha\ da\ matriz\ ma\ *) 
			
            ma[\([\((NN - 1)\)*6 + 4, \((NN - 1)\)*6 + j]\)] = 
              sm\ sn\ mij[\([m, n, NN, 3, j]\)] chs[\([m, n, NN, j]\)]; 
             (*\ F*) 
			
            ma[\([\((NN - 1)\)*6 + 4, \((NN - 1)\)*6 + j + 3]\)] = 
              sm\ sn\ mij[\([m, n, NN, 3, j]\)] shs[\([m, n, NN, j]\)]; 
             (*\ G*) 
			
            ma[\([\((NN - 1)\)*6 + 5, \((NN - 1)\)*6 + j]\)] = 
              cm\ sn\ c1[\([8, NN]\)] al[\([m, n, NN, j]\)] 
                \((mk[\([m, n, NN, j]\)] + pm\ rk[\([m, n, NN, j]\)])\) 
                shs[\([m, n, NN, j]\)]; \  (*\ F*) 
			
            ma[\([\((NN - 1)\)*6 + 5, \((NN - 1)\)*6 + j + 3]\)] = 
              cm\ sn\ c1[\([8, NN]\)] 
                \((mk[\([m, n, NN, j]\)] + pm\ rk[\([m, n, NN, j]\)])\) 
                chs[\([m, n, NN, j]\)]; \ \ \ \ \ \ \ \ \  (*\ G*) 
			
            ma[\([\((NN - 1)\)*6 + 6, \((NN - 1)\)*6 + j]\)] = 
              sm\ cn\ c1[\([7, NN]\)] al[\([m, n, NN, j]\)] 
                \((mk[\([m, n, NN, j]\)]\ lk[\([m, n, NN, j]\)] + 
                    pn\ rk[\([m, n, NN, j]\)])\) shs[\([m, n, NN, j]\)]; \ 
             (*\ F*) 
			
            ma[\([\((NN - 1)\)*6 + 6, \((NN - 1)\)*6 + j + 3]\)] = 
              sm\ cn\ c1[\([7, NN]\)] 
                \((mk[\([m, n, NN, j]\)]\ lk[\([m, n, NN, j]\)] + 
                    pn\ rk[\([m, n, NN, j]\)])\) chs[\([m, n, NN, j]\)], 
            \ \ \ \ \ \  (*\ G*) 
		{j, 3}]
		
		, 
          Print["d ", NN]	]; \ \ \ \ 
         (*\ fim\ do\ if\ caso\ lamina\ ortotropica.\ Agora\ lamina\ 
            isotropica\ *) 

							\  (*\ iso\ iso\ 2\ *) 
	
        If[iso[\([1]\)] == "iso", \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ ISO\ ISO\ 2\ sup.inferior\ do\ laminado\ *) 
\ \ \ 
          kk = \((NN - 1)\)*6; 
	
          Do[
	\ \ 
            ma[\([kk + 1, jj]\)] = 
              sm\ sn\ \((
                  c1[\([1, 1]\)]\ wl[\([m, n, 1, 1, jj]\)] - 

                    								
                    c1[\([2, 1]\)] 
                      \((pm\ uk[\([m, n, 1, 1, jj]\)] + 
                          pn\ vk[\([m, n, 1, 1, jj]\)])\))\); \ 
			
            ma[\([kk + 2, jj]\)] = 
              cm\ sn\ \((
                    \((c1[\([1, 1]\)] - c1[\([2, 1]\)])\) 
                      \((ul[\([m, n, 1, 1, jj]\)] + 
                          pm\ wk[\([m, n, 1, 1, jj]\)])\))\)/2; \ 
             (*tauxz\ *) 
			
            ma[\([kk + 3, jj]\)] = 
              sm\ cn\ \((
                    \((c1[\([1, 1]\)] - c1[\([2, 1]\)])\) 
                      \((vl[\([m, n, 1, 1, jj]\)] + 
                          pn\ wk[\([m, n, 1, 1, jj]\)])\))\)/2\ 
             (*tauyz\ *) 
			\ , {jj, 6}]
\ \ \ , 
          Print["e ", NN]\ \ ]; \ \ \ \ \ \ \ \ \ 
         (*\ fim\ if\ iso\ *) \ \ \ \ \ \ \ \ \ 
	
	\ 
        If[iso[\([NN]\)] == "iso", \ \ \ \ \ \ \ \ \ \ \ 
           (*\ superficie\ superior\ do\ laminado\ *) 
		\ \ \ \ 
          kk = \((NN - 1)\)*6; 
				
          Do[
\ \ \ 
            ma[\([kk + 4, kk + jj]\)] = 
              sm\ sn\ \((
                  c1[\([1, NN]\)] wl[\([m, n, NN, 2, jj]\)] - 

                    								
                    c1[\([2, NN]\)] 
                      \((pm\ uk[\([m, n, NN, 2, jj]\)] + 
                          pn\ vk[\([m, n, NN, 2, jj]\)])\))\); 
             (*\ sigz\ *) 
			
            ma[\([kk + 5, kk + jj]\)] = 
              cm\ sn\ \((
                    \((c1[\([1, NN]\)] - 
								c1[\([2, NN]\)])\) 
                      \((ul[\([m, n, NN, 2, jj]\)] + 
                          pm\ wk[\([m, n, NN, 2, jj]\)])\))\)/2; 
             (*tauxz\ *) 
			
            ma[\([kk + 6, kk + jj]\)] = 
              sm\ cn\ \((
                    \((c1[\([1, NN]\)] - 
								c1[\([2, NN]\)])\) 
                      \((vl[\([m, n, NN, 2, jj]\)] + 
                          pn\ wk[\([m, n, NN, 2, jj]\)])\))\)/2\ \ \ \ \ 
             (*tauyz\ *) 

\ \ \ \ \ \ \ , {jj, 6}]
\ , 
          Print["f ", NN]	\ ]; \ \ \ \ \ \ \ \ \ 
         (*\ fim\ if\ iso\ *) \ \ \ \ 

		
		
        vf[\([6  NN - 2]\)] = qmn[\([m, n]\)] sm\ sn; \ \ \ \ \ \ \ 
         (*\ VETOR\ FORCA\ sup.super.\ *) 

         (*	\ vf[\([6  NN - 5]\)] = \(-qmn[\([m, n]\)]\) sm\ sn/2; \ \ 
           (*\ \ superf.inferior\ \ *) \ \ \ \ \ \ \ \ \ \ \ *) 

		\ 
         (*\ \(--
                \(--\(--
                    \(--\(--
                        \(--\(--
                            \(--\(--
                                \(--
													VALORES
                                  \)\)\)\)\)\)\)\)\)\)\ NA\ SUP.INFERIOR\ DA\ 
              LAM.k + 1\ *) 
\ \ 
        Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ percorre\ as\ laminas\ *) 
		
          If[iso[\([k + 1]\)] == "orto", 
\ \ \ \ 
            Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
               (*\ corre\ j = 1, 2, 3\ da\ linha\ da\ matriz\ \(ma.\)\ *) 

              		\ \ ma[\([\((k - 1)\) 6 + 1, k*6 + j]\)] = 
                \(-cm\)\ sn\ chi[\([m, n, k + 1, j]\)]; \ \ \  (*u\ F\ *) 

              			\ ma[\([\((k - 1)\) 6 + 1, k*6 + j + 3]\)] = 
                \(-cm\)\ sn\ shi[\([m, n, k + 1, j]\)];  (*u\ G*) 
			\ 
              ma[\([\((k - 1)\) 6 + 2, k*6 + j]\)] = 
                \(-lk[\([m, n, k + 1, j]\)]\) sm\ cn\ 
                  chi[\([m, n, k + 1, j]\)]; \ \ \  (*v\ F\ *) 
			\ 
              ma[\([\((k - 1)\) 6 + 2, k*6 + j + 3]\)] = 
                \(-lk[\([m, n, k + 1, j]\)]\) sm\ cn\ 
                  shi[\([m, n, k + 1, j]\)];  (*v\ G*) 
			
              ma[\([\((k - 1)\) 6 + 3, k*6 + j]\)] = 
                \(-rk[\([m, n, k + 1, j]\)]\) 
                  al[\([
									m, n, k + 1, j]\)] sm\ sn\ 
                  shi[\([m, n, k + 1, j]\)]; \ \ \  (*w\ F\ *) 
		\ \ 
              ma[\([\((k - 1)\) 6 + 3, k*6 + j + 3]\)] = \ 
                \(-rk[\([m, n, k + 1, j]\)]\) sm\ sn\ 
                  chi[\([m, n, k + 1, j]\)];  (*w\ G*) 
			
					\ 
               (*\ Tensoes\ sigz, \ tauxz, \ tauyz\ *) 
			
              ma[\([\((k - 1)\) 6 + 4, k*6 + j]\)] = 
                \(-sm\)\ sn\ mij[\([m, n, k + 1, 3, j]\)] 
                  chi[\([m, n, k + 1, j]\)];  (*\ F*) 
			
              ma[\([\((k - 1)\) 6 + 4, k*6 + j + 3]\)] = 
                \(-sm\)\ sn\ mij[\([m, n, k + 1, 3, j]\)] 
                  shi[\([m, n, k + 1, j]\)];  (*\ G*) 
			
              ma[\([\((k - 1)\) 6 + 5, k*6 + j]\)] = 
                \(-cm\)\ sn\ c1[\([8, k + 1]\)] al[\([m, n, k + 1, j]\)] 
                  \((mk[\([m, n, k + 1, j]\)] + pm\ rk[\([m, n, k + 1, j]\)])
                    \) shi[\([m, n, k + 1, j]\)]; \ \ \ \ \ \ \ \ \ \ \ \ \ 
               (*\ F*) 
			
              ma[\([\((k - 1)\) 6 + 5, k*6 + j + 3]\)] = 
                \(-cm\)\ sn\ c1[\([8, k + 1]\)] 
                  \((mk[\([m, n, k + 1, j]\)] + 
									
                      pm\ rk[\([m, n, k + 1, j]\)])\) 
                  chi[\([m, n, k + 1, j]\)]; \  (*\ G*) 
			
              ma[\([\((k - 1)\) 6 + 6, k*6 + j]\)] = 
                \(-sm\)\ cn\ c1[\([7, k + 1]\)] al[\([m, n, k + 1, j]\)] 
                  \((mk[\([m, n, k + 1, j]\)]\ lk[\([m, n, k + 1, j]\)] + 
                      pn\ rk[\([m, n, k + 1, j]\)])\) 
                  shi[\([m, n, k + 1, j]\)]; \ \ \  (*\ F*) 
			
              ma[\([\((k - 1)\) 6 + 6, k*6 + j + 3]\)] = 
                \(-sm\)\ cn\ c1[\([7, k + 1]\)] 
                  \((mk[\([m, n, k + 1, j]\)]\ lk[\([m, n, k + 1, j]\)] + 
                      pn\ rk[\([m, n, k + 1, j]\)])\) 
                  chi[\([m, n, k + 1, j]\)], \ \  (*\ G*) 
			

              					{j, 3}]
	, Print["g ", NN]]; \ \ \ \ 
           (*\ fim\ do\ if\ caso\ lamina\ ortotropica.\ Agora\ lamina\ 
              isotropica\ *) 
							
							\ 
           (*\ ISO\ ISO\ 3\ sup.inferior\ da\ lam\ k + 1\ *) 
	
          If[iso[\([k + 1]\)] == "iso", \ 
             (*\ ISO\ ISO\ 3\ sup.inferior\ da\ lam\ k + 1\ *) 
				
            kk = \((k - 1)\)\ 6; 
			
            Do[
			
              ma[\([kk + 1, 6  k + jj]\)] = 
                \(-cm\)\ sn\ uk[\([m, n, k + 1, 1, jj]\)]; 
              \ \ \ \ \ \ \ \ \ \  (*u\ \ *) 
			
              ma[\([kk + 2, 6  k + jj]\)] = 
                \(-sm\)\ cn\ vk[\([m, n, k + 1, 1, jj]\)]; 
              \ \ \ \ \ \ \ \ \ \  (*v\ \ *) 
			
              ma[\([kk + 3, 6  k + jj]\)] = 
                \(-sm\)\ sn\ wk[\([m, n, k + 1, 1, jj]\)]; 
              \ \ \ \ \ \ \ \ \ \  (*w\ \ *) 
			
              ma[\([kk + 4, 6  k + jj]\)] = 
                \(-sm\)\ sn\ 
                  \((c1[\([1, k + 1]\)] wl[\([m, n, k + 1, 1, jj]\)] - 
                      c1[\([2, k + 1]\)] 
                        \((pm\ uk[\([m, n, k + 1, 1, jj]\)] + 
                            pn\ vk[\([m, n, k + 1, 1, jj]\)])\))\); 
               (*\ sigz\ *) 
			
              ma[\([kk + 5, 6  k + jj]\)] = 
                \(-cm\)\ sn\ 
                  \((\((c1[\([1, k + 1]\)] - c1[\([2, k + 1]\)])\) 
                        \((ul[\([m, n, k + 1, 1, jj]\)] + 

                            															
                            pm\ wk[\([m, n, k + 1, 1, jj]\)])\))\)/2; 
              \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
               (*tauxz\ *) 
		
              ma[\([kk + 6, 6  k + jj]\)] = 
                \(-sm\)\ cn\ 
                  \((\((c1[\([1, k + 1]\)] - c1[\([2, k + 1]\)])\) 
                        \((vl[\([m, n, k + 1, 1, jj]\)] + 

                            														
                            pn\ wk[\([m, n, k + 1, 1, jj]\)])\))\)/2
                    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*tauyz*) 
	\ \ \ \ \ \ \ \ \ \ \ \ 
				\ \ , {jj, 6}]
              \ \ \ \ \  (*\ fim\ do\ do\ jj\ *) 	
	, 
            Print["h ", NN]], \ \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ fim\ do\ if\ caso\ lamina\ \(isotropica.\)\ \ *) 	

          		{k, NN - 1}]; 
	
		\ 
         (*\ SOLUCAO\ para\ F\ e\ G\ em\ cada\ lamina, \ 
          cada\ harmonico\ \ \ \ \ *) 
		vs = LinearSolve[ma, vf]; 
        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 

        		Do[Do[
				
            fg[\([m, n, k, j]\)] = vs[\([\((k - 1)\) 6 + j]\)], 
            \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ F\ e\ G\ *) 

            					{j, 6}], {k, NN}]; 


         (*\ \(--
            \(--\(--\(--
                  \(--\(--
                      \(--\(--
                          \(--\(--
                              \(--\(--
                                  \(--\(--
                                      \(--\(--
                                        \(--\(--
                                        \(--\(--\(--
                                        \(--\(--\(--\(--\(--\(--\(--
                                        \(--\(--\(--\(----
                                        \)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\ *) 
		
         (*\ CALCULO\ DAS\ TENSOES\ E\ DESLOCAMENTOS\ \ \ *) 

         (*\ \(--
            \(--\(--\(--
                  \(--\(--
                      \(--\(--
                          \(--\(--
                              \(--\(--
                                  \(--\(--
                                      \(--\(--
                                        \(--\(--
                                        \(--\(--\(--
                                        \(--\(--\(--\(--\(--\(--
                                        \(--\(--\(--\(--\(----
                                        \)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\ *) 
	
		
\ \ \ \ \ 
        Do[Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ 
             (*\ deslocamentos\ nas\ interfaces\ superiores\ *) 
		\ \ \ \ 
            u[\([k + 1, 2]\)] = 
              u[\([k + 1, 2]\)] + 
                ma[\([\((k - 1)\)*6 + 1, \((k - 1)\)*6 + j]\)]\ 
                  fg[\([m, n, k, j]\)]; 
\ \ 			
            v[\([k + 1, 2]\)] = 
              v[\([k + 1, 2]\)] + 
                ma[\([\((k - 1)\)*6 + 2, \((k - 1)\)*6 + j]\)]\ 
                  fg[\([m, n, k, j]\)]; 
				
            w[\([k + 1, 2]\)] = 
              w[\([k + 1, 2]\)] + 
                ma[\([\((k - 1)\)*6 + 3, \((k - 1)\)*6 + j]\)]\ 
                  fg[\([m, n, k, j]\)]; 
				
			\ \ 
             (*\ tensoes\ transversais\ nas\ interfaces\ superiores\ *) 
	
            sigz[\([k + 1, 2]\)] = \ 
              sigz[\([k + 1, 2]\)] + 
                ma[\([\((k - 1)\)*6 + 4, \((k - 1)\)*6 + j]\)]\ 
                  fg[\([m, n, k, j]\)]; 
	
            tauxz[\([k + 1, 2]\)] = 
              tauxz[\([k + 1, 2]\)] + 
                ma[\([\((k - 1)\)*6 + 5, \((k - 1)\)*6 + j]\)]\ 
                  fg[\([m, n, k, j]\)]; 
	
            tauyz[\([k + 1, 2]\)] = 
              tauyz[\([k + 1, 2]\)] + 
                ma[\([\((k - 1)\)*6 + 6, \((k - 1)\)*6 + j]\)]\ 
                  fg[\([m, n, k, j]\)], 
						
{k, NN - 1}], 
          \ {j, 6}]; 

			

        If[iso[\([1]\)] == "orto", 
	
          Do[\ \ \ \ \ \ \ \ \ 
             (*\ deslocamentos\ na\ superficie\ inferior\ do\ laminado\ *) 

            		\ u[\([1, 2]\)] = 
              u[\([1, 2]\)] + 
                cm\ sn\ chi[\([m, n, 1, j]\)] fg[\([m, n, 1, j]\)]; 
		\ 
            u[\([1, 2]\)] = 
              u[\([1, 2]\)] + 
                cm\ sn\ shi[\([m, n, 1, j]\)] fg[\([m, n, 1, j + 3]\)]; 

            		\ v[\([1, 2]\)] = 
              v[\([1, 2]\)] + 
                sm\ cn\ lk[\([m, n, 1, j]\)] chi[\([m, n, 1, j]\)] 
                  fg[\([m, n, 1, j]\)]; 
		
            v[\([1, 2]\)] = 
              v[\([1, 2]\)] + 
                sm\ cn\ lk[\([m, n, 1, j]\)] shi[\([m, n, 1, j]\)] 
                  fg[\([m, n, 1, j + 3]\)]; 
		
            w[\([1, 2]\)] = 
              w[\([1, 2]\)] + 
                sm\ sn\ al[\([m, n, 1, j]\)] rk[\([m, n, 1, j]\)] 
                  shi[\([m, n, 1, j]\)] fg[\([m, n, 1, j]\)]; 
		
            w[\([1, 2]\)] = 
              w[\([1, 2]\)] + 
                sm\ sn\ rk[\([m, n, 1, j]\)]\ chi[\([m, n, 1, j]\)] 
                  fg[\([m, n, 1, j + 3]\)], 

            																				{j, 3}]
	, 
          Print["i1 ", NN]]; \ \ \ \ \ \ \ \ \ \ \ \ 
         (*\ fim\ if\ orto\ \ *) 
	
	
        If[iso[\([NN]\)] == "orto", 
	
          Do[\ \ \ 						
             (*\ deslocamentos\ na\ superficie\ superior\ do\ laminado\ *) 

            		\ u[\([NN + 1, 2]\)] = 
              u[\([NN + 1, 2]\)] + 
                cm\ sn\ chs[\([m, n, NN, j]\)] fg[\([m, n, NN, j]\)]; 
		\ 
            u[\([NN + 1, 2]\)] = 
              u[\([NN + 1, 2]\)] + 
                cm\ sn\ shs[\([m, n, NN, j]\)] fg[\([m, n, NN, j + 3]\)]; 

            		\ v[\([NN + 1, 2]\)] = 
              v[\([NN + 1, 2]\)] + 
                sm\ cn\ lk[\([m, n, NN, j]\)] chs[\([m, n, NN, j]\)] 
                  fg[\([m, n, NN, j]\)]; 
\ 		
            v[\([NN + 1, 2]\)] = 
              v[\([NN + 1, 2]\)] + 
                sm\ cn\ lk[\([m, n, NN, j]\)] shs[\([m, n, NN, j]\)] 
                  fg[\([m, n, NN, j + 3]\)]; 
\ \ \ 
            w[\([NN + 1, 2]\)] = 
              w[\([NN + 1, 2]\)] + 
                sm\ sn\ al[\([m, n, NN, j]\)] rk[\([m, n, NN, j]\)]\ 
                  shs[\([m, n, NN, j]\)] fg[\([m, n, NN, j]\)]; 
\ 		
            w[\([NN + 1, 2]\)] = 
              w[\([NN + 1, 2]\)] + 
                sm\ sn\ rk[\([m, n, NN, j]\)]\ chs[\([m, n, NN, j]\)] 
                  fg[\([m, n, NN, j + 3]\)], 

            																				{j, 3}]
	, 
          Print["i ", NN]]; \ \ \ \ \ \ \ \ \ \ \ \ 
         (*\ fim\ if\ orto\ \ *) 
				
							\ 
         (*\ ISO\ ISO\ 4\ deslocamentos\ na\ superficie\ inf\ do\ laminado
            \ *) 
If[iso[\([1]\)] == "iso", \ \ \ \ 
           (*\ ISO\ ISO\ 4\ deslocamentos\ na\ superficie\ inf\ do\ laminado
              \ *) 
			
          Do[
			\ 
            u[\([1, 2]\)] = 
              u[\([1, 2]\)] + 
                cm\ sn\ uk[\([m, n, 1, 1, jj]\)] fg[\([m, n, 1, jj]\)]; 
            \ \ \ \ \ \ \ \ \  (*u\ \ *) 
			\ 
            v[\([1, 2]\)] = 
              v[\([1, 2]\)] + 
                sm\ cn\ vk[\([m, n, 1, 1, jj]\)] fg[\([m, n, 1, jj]\)]; 
            \ \ \ \ \ \ \ \ \  (*\ v\ \ *) 
			\ 
            w[\([1, 2]\)] = 
              w[\([1, 2]\)] + 
                sm\ sn\ wk[\([m, n, 1, 1, jj]\)] 
                  fg[\([m, n, 1, jj]\)]\ \ \ \ \ \ \ \ \  (*w\ \ *) 

            		\ , {jj, 6}]\ , Print["j ", NN]]; \ \ \ \ \ \ \ \ \ \ \ 
         (*\ fim\ if\ iso\ \ *) 
	

        If[iso[\([NN]\)] == "iso", \ \ \ \ 
           (*\ deslocamentos\ na\ superficie\ superior\ do\ laminado\ *) 

          			Do[
			\ 
            u[\([NN + 1, 2]\)] = 
              u[\([NN + 1, 2]\)] + 
                cm\ sn\ uk[\([m, n, NN, 2, jj]\)] fg[\([m, n, NN, jj]\)]; 
            \ \ \ \ \ \ \ \ \  (*u\ \ *) 
			\ 
            v[\([NN + 1, 2]\)] = 
              v[\([NN + 1, 2]\)] + 
                sm\ cn\ vk[\([m, n, NN, 2, jj]\)] fg[\([m, n, NN, jj]\)]; 
            \ \ \ \ \ \ \ \ \  (*\ v\ \ *) 
			\ 
            w[\([NN + 1, 2]\)] = 
              w[\([NN + 1, 2]\)] + 
                sm\ sn\ wk[\([m, n, NN, 2, jj]\)] 
                  fg[\([m, n, NN, jj]\)]\ \ \ \ \ \ \ \ \  (*w\ \ *) 

            		\ , {jj, 6}]\ , Print["k ", NN]]; \ \ \ \ \ \ \ \ \ \ \ 
         (*\ fim\ if\ iso\ \ *) 


		
        Do[\ \ 		
           (*\ tensoes\ transversais\ na\ superficie\ inferior\ do\ laminado
              \ *) 
	sigz[\([1, 2]\)] = \ 
            sigz[\([1, 2]\)] + 
              ma[\([\((NN - 1)\)*6 + 1, j]\)]\ fg[\([m, n, 1, j]\)]; 
	
          tauxz[\([1, 2]\)] = \ 
            tauxz[\([1, 2]\)] + 
              ma[\([\((NN - 1)\)*6 + 2, j]\)]\ fg[\([m, n, 1, j]\)]; 
	
          tauyz[\([1, 2]\)] = \ 
            tauyz[\([1, 2]\)] + 
              ma[\([\((NN - 1)\)*6 + 3, j]\)]\ fg[\([m, n, 1, j]\)]; 
			
          
						
           (*\ tensoes\ transversais\ na\ superficie\ superior\ do\ laminado
              \ *) 
	sigz[\([NN + 1, 2]\)]\ \  = \ 
            sigz[\([NN + 1, 2]\)]\ \ \  + 
              ma[\([\((NN - 1)\)*6 + 4, \((NN - 1)\) 6 + j]\)]\ 
                fg[\([m, n, NN, j]\)]; 
	
          tauxz[\([NN + 1, 2]\)] = \ \ 
            tauxz[\([NN + 1, 2]\)] + 
              ma[\([\((NN - 1)\)*6 + 5, \((NN - 1)\) 6 + j]\)]\ 
                fg[\([m, n, NN, j]\)]; 
	
          tauyz[\([NN + 1, 2]\)] = \ 
            tauyz[\([NN + 1, 2]\)] + 
              ma[\([\((NN - 1)\)*6 + 6, \((NN - 1)\) 6 + j]\)]\ 
                fg[\([m, n, NN, j]\)], 


          																{j, 6}]; 
		
				
         (*\ \(--\(--\(--\(--\(-\ Tensoes\)\)\)\)\)\ coplanares\ na\ 
              superficie\ superior\ das\ \(laminas\ --\) - *) 

        Do[If[iso[\([k]\)] == "orto", 
            Do[

	\ 
              sigx[\([2  k, 2]\)] = 
                sigx[\([2  k, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 1, j]\)] chs[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j]\)]; 		
\ 
              sigx[\([2  k, 2]\)] = 
                sigx[\([2  k, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 1, j]\)] shs[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j + 3]\)]; 
	\ \ 
              sigy[\([2  k, 2]\)] = 
                sigy[\([2  k, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 2, j]\)] chs[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j]\)]; 
	\ \ 
              sigy[\([2  k, 2]\)] = 
                sigy[\([2  k, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 2, j]\)] shs[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j + 3]\)]; 

              tauxy[\([2  k, 2]\)] = 
                tauxy[\([2  k, 2]\)] + 
                  cm\ cn\ c1[\([9, k]\)]\ 
                    \((pn + 
									pm\ lk[\([m, n, k, j]\)])\) 
                    chs[\([m, n, k, j]\)]\ fg[\([m, n, k, j]\)]; 
	\ 
              tauxy[\([2  k, 2]\)] = 
                tauxy[\([2  k, 2]\)] + 
                  cm\ cn\ c1[\([9, k]\)]\ 
                    \((pn + 
									pm\ lk[\([m, n, k, j]\)])\) 
                    shs[\([m, n, k, j]\)]\ fg[\([m, n, k, j + 3]\)]; 

	

              		 (*\ \(--\(--\(--\(--\(-\ Tensoes\)\)\)\)\)\ coplanares\ na
                    \ superficie\ inferior\ das\ \(laminas\ --\) - *) 
	\ 
              sigx[\([2  k - 1, 2]\)] = 
                sigx[\([2  k - 1, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 1, j]\)] chi[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j]\)]; 
	\ 
              sigx[\([2  k - 1, 2]\)] = 
                sigx[\([2  k - 1, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 1, j]\)] shi[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j + 3]\)]; 
	\ \ 
              sigy[\([2  k - 1, 2]\)] = 
                sigy[\([2  k - 1, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 2, j]\)] chi[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j]\)]; 
	\ \ 
              sigy[\([2  k - 1, 2]\)] = 
                sigy[\([2  k - 1, 2]\)] + 
                  sm\ sn\ mij[\([m, n, k, 2, j]\)] shi[\([m, n, k, j]\)]\ 
                    fg[\([m, n, k, j + 3]\)]; 
	\ 
              tauxy[\([2  k - 1, 2]\)] = 
                tauxy[\([2  k - 1, 2]\)] + 
                  cm\ cn\ c1[\([9, k]\)]\ \((pn + pm\ lk[\([m, n, k, j]\)])\) 
                    chi[\([m, n, k, j]\)]\ fg[\([m, n, k, j]\)]; 
	\ 
              tauxy[\([2  k - 1, 2]\)] = 
                tauxy[\([2  k - 1, 2]\)] + 
                  cm\ cn\ c1[\([9, k]\)]\ 
                    \((pn + 
							pm\ lk[\([m, n, k, j]\)])\) 
                    shi[\([m, n, k, j]\)]\ fg[\([m, n, k, j + 3]\)]; 
	

              			Print["fim ", NN], {j, 3}], Print["fim ", NN]\ ]; 
          \ \ \ \  (*\ fim\ if\ orto\ \ \ \ \ \ \ \ \ *) 
		

          							\  (*\ iso\ iso\ 5\ *) 
	
          \(If[iso[\([k]\)] == "iso", \ \ \ 
\ \ \ kk = \((NN - 1)\)*6; 
            
	Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
               (*\ ISO\ ISO\ 5\ \ Tensoes\ coplanares\ na\ superficie\ sup\ 
                  das\ laminas\ *) 
	\ 
              sigx[\([2  k, 2]\)] = 
                sigx[\([2  k, 2]\)] + 
                  sm\ sn\ \((
                      \(-pm\)\ c1[\([1, k]\)]\ uk[\([m, n, k, 2, jj]\)] + 

                        			c1[\([2, k]\)] 
                          \((\(-pn\)\ vk[\([m, n, k, 2, jj]\)] + 
                              wl[\([m, n, k, 2, jj]\)])\))\) 
                    fg[\([m, n, k, jj]\)]; \ \  (*\ sigx\ *) 
	
              sigy[\([2  k, 2]\)] = 
                sigy[\([2  k, 2]\)] + 
                  sm\ sn\ \((
                      \(-pn\)\ \ c1[\([1, k]\)]\ vk[\([m, n, k, 2, jj]\)] + 

                        			c1[\([2, k]\)] 
                          \((\(-pm\)\ uk[\([m, n, k, 2, jj]\)] + 
                              wl[\([m, n, k, 2, jj]\)])\))\) 
                    fg[\([m, n, k, jj]\)];  (*\ sigy\ *) 
	
              tauxy[\([2  k, 2]\)] = 
                tauxy[\([2  k, 2]\)] + 
                  cm\ cn\ \((
                      \((\ \ c1[\([1, k]\)]\  - 
		c1[\([2, k]\)])\) 
                        \((pn\ uk[\([m, n, k, 2, jj]\)] + 
                            pm\ vk[\([m, n, k, 2, jj]\)])\))\) 
                    fg[\([m, n, k, jj]\)]\ /2;  (*\ tauxy\ *) 
		
		\ 
               (*\ \ Tensoes\ coplanares\ na\ superficie\ inferior\ das\ 
                  laminas\ *) 
	\ 
              sigx[\([2  k - 1, 2]\)] = 
                sigx[\([2  k - 1, 2]\)] + 
                  sm\ sn\ \((
                      \(-pm\)\ c1[\([1, k]\)]\ uk[\([m, n, k, 1, jj]\)] + 

                        		c1[\([2, k]\)] 
                          \((\(-pn\)\ vk[\([m, n, k, 1, jj]\)] + 
                              wl[\([m, n, k, 1, jj]\)])\))\) 
                    fg[\([m, n, k, jj]\)]; \ \ \  (*\ sigx\ *) 
	
              sigy[\([2  k - 1, 2]\)] = 
                sigy[\([2  k - 1, 2]\)] + 
                  sm\ sn\ \((
                      \(-pn\)\ \ c1[\([1, k]\)]\ vk[\([m, n, k, 1, jj]\)] + 

                        		c1[\([2, k]\)] 
                          \((\(-pm\)\ uk[\([m, n, k, 1, jj]\)] + 
                              wl[\([m, n, k, 1, jj]\)])\))\) 
                    fg[\([m, n, k, jj]\)];  (*\ sigy\ *) 
	
              tauxy[\([2  k - 1, 2]\)] = 
                tauxy[\([2  k - 1, 2]\)] + 
                  cm\ cn\ \((
                      \((\ \ c1[\([1, k]\)]\  - 
		c1[\([2, k]\)])\) 
                        \((pn\ uk[\([m, n, k, 1, jj]\)] + 
                            pm\ vk[\([m, n, k, 1, jj]\)])\))\) 
                    fg[\([m, n, k, jj]\)]\ /2\ \ \  (*\ tauxy\ *) 


              			\ , {jj, 6}]
\ \ \ , Print["e ", NN]\ \ ]; 
          \)\ \ \ \ \ \ \ \ \  (*\ fim\ if\ iso\ *) \ \ \ \ \ \ \ \ \ 
	
	
          
	, {k, NN}]; 
		Print["fim ", NN], 
		{m, mh}], {n, 
        nh}]; \)
\)\)

\(	\ \ TableForm[ma]\ \)\(\ \ \ 	TableForm[vs]\)\(TableForm[fg]
 (*\ TableForm[vf]

      \ \ \ TableForm[fg]\ \ \ *) 
	\)

\(\(sigxN = sigx  H\^2\/\(q0\ a\^2\); \ \ \ \ \ 
    sigyN = sigy  H\^2\/\(q0\ a\^2\); \ \ \ \ \ 
    tauxyN = tauxy  H\^2\/\(q0\ a\^2\); \ \ \ \ \ 

    tauxzN = tauxz  H\/\(q0\ a\); \ \ \ \ \ tauyzN = tauyz  H\/\(q0\ a\); 
    \ \ \ \ \ 
uN = u \( e2[\([1]\)] H\^2\)\/\(q0\ a\^3\); 
    \ \ \ \ \ \ \ \ \ \ vN = v \( e2[\([1]\)] H\^2\)\/\(q0\ a\^3\); 
    \ \ \ \ \ \ \ \ \ \ wN = w \( 100  e2[\([1]\)] H\^3\)\/\(q0\ a\^4\); 
    \ \ \ \ \ \ \ \ \ \ \)\)

\(\( (*\ 
      COLOCAR\ COORDENADAS\ Z = Z\ na\ primeira\ coluna\ dos\ vetores\ u, v, 
      w, sigx, etc\ *) 

    Do[sigx[\([2  k - 1, 1]\)] = z[\([k]\)]\ ; \ \ \ \ \ \ \ 
      sigx[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 
	\ \ \ \ 
      sigy[\([2  k - 1, 1]\)] = z[\([k]\)]\ ; \ \ \ \ \ \ \ 
      sigy[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 
\ 	\ \ \ 
      tauxy[\([2  k - 1, 1]\)] = z[\([k]\)]\ ; \ \ \ \ \ \ \ 
      tauxy[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 
	
	\ \ \ \ 
      tauxz[\([k, 1]\)] = z[\([k]\)]\ ; 
	\ \ \ \ \ 
      tauyz[\([k, 1]\)] = z[\([k]\)]\ ; 
	\ \ \ \ \ \ 
      sigz[\([k, 1]\)] = z[\([k]\)]\ ; \ 
	\ \ \ \ \ 
      u[\([k, 1]\)] = z[\([k]\)]; 
	\ \ \ \ \ v[\([k, 1]\)] = z[\([k]\)]; 

      	\ \ \ \ \ w[\([k, 1]\)] = z[\([k]\)], 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {k, NN}]; 
\ \ \ \ \ \ 
    u[\([NN + 1, 1]\)] = z[\([NN + 1]\)]; 
	\ \ \ \ \ 
    v[\([NN + 1, 1]\)] = z[\([NN + 1]\)]; 
	\ \ \ \ \ 
    w[\([NN + 1, 1]\)] = z[\([NN + 1]\)]; 
\ \ \ \ \ \ \ 
    tauxz[\([NN + 1, 1]\)] = z[\([NN + 1]\)]; 
	\ \ \ \ \ 
    tauyz[\([NN + 1, 1]\)] = z[\([NN + 1]\)]; 
	\ \ \ \ \ 
    sigz[\([NN + 1, 1]\)] = z[\([NN + 1]\)]; 

     (*\ COLOCAR\ COORDENADAS\ Z = Z/H\ na\ primeir`coluna\ dos\ vetores\ uN, 
      vN, wN, sigxN, etc\ *) 

    Do[sigxN[\([2  k - 1, 1]\)] = z[\([k]\)]/H\ ; \ \ \ \ \ \ \ 
      sigxN[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]/H; 
	\ \ \ \ 
      sigyN[\([2  k - 1, 1]\)] = z[\([k]\)]/H\ ; \ \ \ \ \ 
      Ð\ \ sigyN[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]/H; 
      
\ 	\ \ \ tauxyN[\([2  k - 1, 1]\)] = z[\([k]\)]/H\ ; \ \ \ \ \ \ \ 
      tauxyN[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]/H; 
	
	\ \ \ \ 
      tauxzN[\([k, 1]\)] = z[\([k]\)]/H\ ; 
	\ \ \ \ \ 
      tauyzN[\([k, 1]\)] = z[\([k]\)]/H\ ; 
	\ \ \ \ \ \ 
      sigzN[\([k, 1]\)] = z[\([k]\)]/H\ ; \ 
	\ \ \ \ \ 
      uN[\([k, 1]\)] = z[\([k]\)]/H; 
	\ \ \ \ \ 
      vN[\([k, 1]\)] = z[\([k]\)]/H; 
	\ \ \ \ \ 
      wN[\([k, 1]\)] = z[\([k]\)]/H, 

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {k, NN}]; 
\ \ \ \ \ \ 
    uN[\([NN + 1, 1]\)] = z[\([NN + 1]\)]/H; 
	\ \ \ \ \ 
    vN[\([NN + 1, 1]\)] = z[\([NN + 1]\)]/H; 
	\ \ \ \ \ 
    wN[\([NN + 1, 1]\)] = z[\([NN + 1]\)]/H; 
\ \ \ \ \ \ \ 
    tauxzN[\([NN + 1, 1]\)] = z[\([NN + 1]\)]/H; 
	\ \ \ \ \ 
    tauyzN[\([NN + 1, 1]\)] = z[\([NN + 1]\)]/H; 
	\ \ \ \ \ 
    sigzN[\([NN + 1, 1]\)] = z[\([NN + 1]\)]/H; \)\)

ListPlot[wN,PlotJoined -> True,Frame -> True,GridLines -> Automatic]TableForm[N[wN, 15]]

ListPlot[tauyzN,PlotJoined -> True,Frame -> True,GridLines -> Automatic]TableForm[tauyzN]tauyzN

ListPlot[sigxN,PlotJoined -> True,Frame -> True,GridLines -> Automatic]TableForm[sigxN // N]

ListPlot[sigyN,PlotJoined -> True,Frame -> True,GridLines -> Automatic]TableForm[sigyN // N]