\(\( (*\ \ DADOS\ *) 
NN\  = 3\ ; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Numero\ de\ laminas\ *) 
teta = {0, 90, 0} Pi/180; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Angulos\ das\ laminas.\ A\ lamina\ 1\ e' inferior, \ 
      com\ cota\ z\ \(negativa.\)\ *) 
e2 = {\ 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}; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Coef.Poisson\ na\ direcao\ 12\ de\ cada\ lamina\ *) 

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

    e3 = e2; 
nu13 = {0.25, \ 0.25, \ 0.25}; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Coef.Poisson\ na\ direcao\ 13\ de\ cada\ lamina\ *) 

    nu23 = {0.25, \ 0.25, \ 0.25}; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Coef.Poisson\ na\ direcao\ 23\ de\ cada\ lamina\ *) 

    kapa\  = \ 5/6\ \ ; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (*\ fator\ de\ cisalhamento\ *) 
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 = 120; 
b = 360; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Lados\ da\ placa.\ a\ na\ direcao\ x\ *) 
xc = \ \ 1.001\ a/2; 
    \ \ \ \ \  (*\ Ponto\ para\ o\ calculo\ das\ tensoes\ \(coplanares.\)\ *) 
      
yc = \ \ 1.001\ b/2; \ \ \ \ \ \ \ \ \ \ \ \ \ 

     (*\ Em\ algumas\ podicoes, \ 
      como\ ao\ longo\ das\ linhas\ centrais\ x = \(a/2\ e\ y = b/2\), \ 
      a\ matriz\ associada\ e'\ singular.\ Recomenda - 
        se\ entao\ fazer\ os\ calculos\ num\ ponto\ proximo, \ por\ exemplo, 
      \ xc = \ \ \(1.001\ a/2\ em\ lugar\ de\ xc = a/2.\)\ *) 
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)

\(\( (*\ 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}]\)\)

\(\(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]; 
			
          fi = ArcCos[\(\(-\@27\) fk\)\/\(2 \((\(-ddk\))\)\^\(3/2\)\)]; 

          			
			
          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}], 			

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

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

fkddkfiTableForm[gk]TableForm[mij]

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

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

      	\ \ \ PARA\ CADA\ LAMINA\ EM\ CADA\ HARMONICO\ *) 

    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[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
       (*\ VALORES\ NA\ SUP.SUPERIOR\ DA\ LAM.k\ *) 
\ \ 
      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\ superiores\ *) 
\ \ \ \ 
          Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
             (*\ corre\ j = 1, 2, 3\ da\ linha\ da\ matriz\ ma\ *) 

            				\ 
            ma[\([\((k - 1)\)*6 + 1, \((k - 1)\)*6 + j]\)] = 
              cm\ sn\ chs[\([m, n, k, j]\)]; \ \ \  (*u\ F\ *) 
				\ 
            ma[\([\((k - 1)\)*6 + 1, \((k - 1)\)*6 + j + 3]\)] = 
              cm\ sn\ shs[\([m, n, k, j]\)];  (*u\ G*) 
			\ 
            ma[\([\((k - 1)\)*6 + 2, \((k - 1)\)*6 + j]\)] = 
              lk[\([m, n, k, j]\)] sm\ cn\ chs[\([m, n, k, j]\)]; \ \ \ 
             (*v\ F\ *) 
			\ 
            ma[\([\((k - 1)\)*6 + 2, \((k - 1)\)*6 + j + 3]\)] = 
              lk[\([m, n, k, j]\)] sm\ cn\ shs[\([m, n, k, j]\)];  (*v\ G*) 

            			ma[\([\((k - 1)\)*6 + 3, \((k - 1)\)*6 + j]\)] = 
              rk[\([m, n, k, j]\)] al[\([m, n, k, j]\)] sm\ sn\ 
                shs[\([m, n, k, j]\)]; \ \ \  (*w\ F\ *) 
		\ \ 
            ma[\([\((k - 1)\)*6 + 3, \((k - 1)\)*6 + j + 3]\)] = 
              rk[\([m, n, k, j]\)] sm\ sn\ chs[\([m, n, k, j]\)]; \ \ \ 
             (*w\ G*) 
			
					\ 
             (*\ Tensoes\ sigz, \ tauxz, \ tauyz, \ interfaces, 
              sup.\ superior\ *) 
			
            ma[\([\((k - 1)\)*6 + 4, \((k - 1)\)*6 + j]\)] = 
              sm\ sn\ mij[\([m, n, k, 3, j]\)] chs[\([m, n, k, j]\)]; 
             (*\ F*) 
			
            ma[\([\((k - 1)\)*6 + 4, \((k - 1)\)*6 + j + 3]\)] = 
              sm\ sn\ mij[\([m, n, k, 3, j]\)] shs[\([m, n, k, j]\)]; 
             (*\ G*) 
			
            ma[\([\((k - 1)\)*6 + 5, \((k - 1)\)*6 + 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[\([\((k - 1)\)*6 + 5, \((k - 1)\)*6 + 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[\([\((k - 1)\)*6 + 6, \((k - 1)\)*6 + 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[\([\((k - 1)\)*6 + 6, \((k - 1)\)*6 + 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}], 
	{k, NN - 1}]; 
		
		
        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*) 
		
				
           (*\ As\ 3\ linhas\ das\ tensoes\ na\ superficie\ superior\ do\ 
              \(lam.\)\ *) 
			
          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}]; 
		
		
        vf[\([6  NN - 2]\)] = qmn[\([m, n]\)] sm\ sn; \ \ \ \ \ \ \ 
         (*\ VETOR\ FORCA\ \ *) 
		
		\ 
         (*\ \(--
                \(--\(--
                    \(--\(--
                        \(--\(--
                            \(--\(--
                                \(--
													VALORES
                                  \)\)\)\)\)\)\)\)\)\)\ NA\ SUP.INFERIOR\ DA\ 
              LAM.k + 1\ *) 
	\ \ 
        Do[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
           (*\ percorre\ as\ laminas\ *) 
\ \ \ \ 
          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}], 
	{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\ *) 

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

          																{j, 3}]; 
		
				
         (*\ \(--\(--\(--\(--\(-\ Tensoes\)\)\)\)\)\ coplanares\ na\ 
              superficie\ superior\ das\ \(laminas\ --\) - *) 
	
        Do[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[\([m, n, k, j + 3]\)], 

            										

            																						{k, NN}], {j, 3}], 

        
		
		{m, mh}], {n, nh}]; \)
 (*\ \ TableForm[ma]\ *) \)\)

\(\(
 (*\ N[ma[\([7, 7]\)], 15]

      TableForm[vf]

      TableForm[vs]

      TableForm[fg]\ 

      TableForm[vf]\ \ \ \ *) 
	\)\)

\(\(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\ primeira\ 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\ ; \ \ \ \ \ \ \ 
      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[w,PlotJoined -> True,Frame -> True,GridLines -> Automatic]TableForm[w]TableForm[wN]

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

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