\(\(\(\ \)\( (*\(\(\(\(\(\(\(\(\(\(\(\(\(\(\(\(\(\(\(\(\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--
																	PROGRAMA\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\ DE\ SOLUCOES\ ANALITICAS\ DE\ PLACAS\ 
	 (*\ LAMINADO\ RETANGULAR\ SIMETRICO\ nao - 
                                        ORTOTROPICO\ angular\ SIMPLESMENTE\ APOIADO\ 
			\ \ SOB\ CARGA\ UNIFORME\ qo\ OU\ CARGA\ SENOIDAL . \ 
                                        D16\ e\ D26\ nao\ nulos . \ 
                                        Tipo\ [\(a/\(-a\)\)/
                                        a] . 
		\ \ \ TEORIA\ DE\ LAMINADO\ \(\(DELGADO\)\(.\)\)\ *) 
	

                                        	
                                        Paulo\ de\ Tarso\ R . \ 
                                        Mendonca, \ \(06/07\)/
                                        2000. 
GRANTE\  - \ \(Grupo\ de\ Analise\ e\ Projeto\ Mecânici\)
\(Departamento\ de\ Engenharia\ Mecanica\  - \ UFSC\)
\(Florianopolis, \ SC\  - \ 
                                        CP\ 476\)
\(www . 
                                        grante . ufsc . br, \ 
                                        mendonca@grante . ufsc . 
                                        br\)
\(\((48)\)\ 3331\) - 9899
	
	\(INSTRUCOES\ DE\ \(USO : 
	
                                        A\)\)\)\()\)\)\ Entre\ com\ os\ dados\ na\ celula\ abaixo\ e\ execute\ o\ frontend\ 
		\ \ \((Kernel -> \(Evaluation -> Evaluate\ Notebook\))\) . 
			
	
                                        B\)\()\)\)\ Role\ para\ a\ ultima\ celula\ e\ escolha\ os\ resultados\ que\ deseja\ listados\ ou\ plotados . 
			\ Os\ comandos\ de\ listar\ e\ plotar\ ja\ estao\ prontos . \ 
                                        Basta\ altera - 
                                        los\ para
	\ \ \ \(o\ dado\ desejado\ e\ executar\ a\ celula\ usando\ Shift + 
                                        Enter . 
				
	
                                        C\)\)\()\)\)\ As\ saidas\ disponiveis\ \(sao : \ deslocamento\ wp\ no\ ponto\), \ wmax\ maximo, \ 
                                      momentos\ fletores\ e\ 
		\ \ \(torcores\ Mx\), My, \ Mxy\ e\ tensoes\ em\ cada\ interface, \ 
                                      nas\ direcoes\ x - y\ e\ 1 - 2, \ 
                                      de\ forma
	\ \ \ \(nao\ normalizada\ e\ normalizada . \ 
                                        Os\ nomes\ sao\ os\ \(seguintes : 
			\ \ wp\)\), \ \ \ wmax, \ \ mx, \ \ my, \ \ mxy, \ \ \ \ sigx, \ \ sigy, \ \ tauxy, \ 
                                      tauxz\ \ \ \ tauxy, \ \ \ \ \(\(->\)\(\ \)\(valores\ nao\ normalizados\)\), \ e
	\ \ \ \ wpN, \ \ wmaxN, \ 
                                      mx, \ \ my, \ \ mxy, \ \ sigxN, \ 
                                      sigyN, \ tauxyN, \ tauyzN, \ 
                                      tauxzN\ \ \ \ \ \  -> \ 
                                        valores\ normalizados . 
					Os\ valores\ acima\ de\ tensoes\ cisalhantes\ são\ obtidos\ pelas\ relacoes
	\ \ \ constitutivas . \ 
                                        Valores\ obtidos\ pela\ integracao\ das\ equacoes\ 
	\ \ \ \(diferenciais\ de\ equilibrio\ \(sao : 
			\ \ \ tauyzi\)\), \ tauxzi, \ tauyziN, \ 
                                      tauxziN . 
				\((E'\ ainda\ feita\ analise\ de\ seguranca, \ conforme\ itens\ H, \ 
                                        I\ e\ J\ abaixo)\)
				

                                        D\)\(	\)\()\)\)\ As\ normalizacoes\ sao\ aquelas\ usadas\ por\ \(Pagano : 
			\ \ \((\ sigxN, \ sigyN, \ 
                                        tauxyN)\)\)\  = \ \((sigx, \ \ sigy, \ \ tauxy)\) H\^2/\((qo\ a\^2)\), 
	\ \ \ \ wpN\  = \ 
                                  100\ D11\ w/\((qo\ a\^4)\), 
	\ \ \ \ \((mxN, \ myN, \ mxyN)\)\  = \ \((mxN, \ myN, \ 
                                      mxyN)\)\ H/\((qo\ a\^3)\); 
	\ \ \ \ \(onde\ E2, \ H, \ a\ e\ qo\ sao\ modulo, \ espessura\ do\ laminado, \ 
                                lado\ da\ placa\ e\ carga, \ 
	\ \ \ \ como\ definidos\ na\ celula\ abaixo . 
				
	
                                    E\)\)\()\)\)\ Os\ valores\ sao\ listados\ e\ plotados\ em\ termos\ das\ coordenadas\ z\ de\ cada\ interface . \ 
	\ \ \ As\ laminas\ sao\ contadas\ no\ sentido\ positivo\ de\ z . \ 
                                A\ origem\ e'\ na\ superficie\ media
	\ \ \ \(do\ laminado . \ 
                                  Os\ valores\ normalizados\ sao\ listados\ e\ plotados\ em\ termos\ de\ zN\), \ 
	\ \ \ definido\ como\ zN\  = \ z/H, \ 
                            variando\ de\  - 
                              1/2\ a\ 1/2. 

	
                                F\)\()\)\)\ Observe\ que\ o\ programa\ so\ funciona\ para\ laminados\ de\ laminas\ ortotropicas
			\ \(alinhadas\ aos\ eixos\ x - y\ ou\ laminas\ isotropicas . 
					
	G\)\)\()\)\) 
                      Pode - se\ entrar\ com\ laminas\ obliquas . \ 
                        
                        As\ matrizes\ de\ material\ 
			serao\ calculadas\ corretamente\ mas\ nos\ calculos\ subsequentes\ os\ termos\ de\ 
		\ \(acoplamento\ sao\ ignorados . 
			
	
                          H\)\)\()\)\)\ A\ analise\ de\ resistencia\ e'\ feita\ nas\ superficies\ inferior\ e\ superior\ de\ 
\ 		\(cada\ lamina\), \ pelos\ metodos\ da\ Tensao\ Maxima\ \((TM)\)\ e\ Tsai - 
                Hill\ \((TH)\) . 
			Caso\ nao\ se\ deseje\ esta\ analise, \ 
              ou\ nao\ se\ disponha\ dos\ \ 
	\ 	\(valores\ de\ resistencia\), \ 
              basta\ colocar\ valores\ bastantes\ altos\ para\ 
		\ \(os\ dados\ de\ resistencia . 
			
	
                    I\)\)\()\)\)\ O\ resultado\ da\ analise\ de\ seguranca\ e'\ o\ FATOR\ DE\ SOLICITACAO
	\ \ \ \(f\ definido\), \ 
          dependendo\ do\ criterio\ \(como : 
			\ Tensao\ \(Maxima : \ 
                    f\)\) = \(Maximo {\((sig1\/Xt\ se\ sig1 > 
                        0\ ou\ sig1\/\(\(\ \)\(Xc\)\)\ se\ sig1 < 
                        0)\), \((sig2\/Yt\ se\ sig2 > 
                        0\ ou\ sig2\/Yc\ se\ sig2 < 0)\), 
                    sig12\/S}
		\ \ Tsai - \(Hill : \ \((sig1\/Xt)\)\^2 - \(\(sig1\ sig2\)\/\(Xt\ Yt\)\) \((sig2\/Yt)\)\^2 + \((sig12\/S)\)\^2\) = 
              f\^2\ caso\ sig1 > 0\ e\ sig2 > 0\), \ 
          etc . 
		
	
            J\)\()\)\)\ O\ coeficiente\ de\ seguranca\ no\ ponto\ \((xc, \ 
          yc)\), \ para\ cada\ criterio, \ 
		\ \ cstm, \ csth, \ 
      e'\ obtido\ por\ 1/
          Max[f]\ dentre\ todas\ as\ interfaces\ de\ laminas . 
			
	
	\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(---\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)*) \)\)\)

\(\( (*\ \ DADOS\ *) 
n\  = \ 3\ ; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Numero\ de\ laminas\ *) 
teta = {\(-45\), 45, \(-45\)} Pi/180; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Angulos\ das\ laminas.\ A\ lamina\ 1\ e' inferior, \ 
      com\ cota\ z\ \(negativa.\)\ *) 

    e1 = {\ \ \ 29504.4, \ 29504.4, 29504.4\ \ }; 
     (*\ Modulo\ na\ direcao\ 1\ de\ cada\ lamina.\ MPa\ *) 

    e2 = {\ \ 2152.68, \ 2152.68, 2152.68}; \ \ \ \ \ \ \ \ 
     (*\ Modulo\ na\ direcao\ 2\ de\ cada\ lamina\ *) 

    g12 = {\ 4500, \ \ 4500, \ \ 4500}; \ \ \ \ \ \ 
     (*\ Modulo\ na\ direcao\ 12\ de\ cada\ lamina\ *) 

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

    h = {10, 10, 10\ }/2; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Espessuras\ das\ laminas.\ mm.\ *) 
mh\  = \ 7; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ numero\ de\ termos\ na\ serie\ em\ x\ *) 
nh\  = \ 7; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ numero\ de\ termos\ na\ serie\ em\ 
        y.\ 
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Convergencia\ lenta.\ 9\ 
        termos\ resultam\ em\ erros\ da\ ordem\ 

      		de\ 4  %\ e\ 13\ termos\ erro\ de\ 2 \( %.\)*) 
qo\  = \ 0.001; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Carga\ distribuida, \ F/Area.\ Se\ mh = \(nh = 1\), \ q0\ e'

      					\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
      o\ modulo\ de\ carga\ senoidal.\ Se\ mh, nh\ diferentes\ de

      	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1, 
      \ carga\ uniforme\ de\ valor\ qo\ *) 
a = 1000; 
b = 1000; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Lados\ da\ placa.\ a\ na\ direcao\ x\ *) 
xc = \ a/2; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Ponto\ para\ o\ calculo\ das\ tensoes\ coplanares, \ 
      deslocamento\ e\ \(momentos.\)*) 
yc = b/2; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ A\ rigem\ dos\ eixos\ e' no\ vertice\ do\ retangulo\ *) 

    Xt = {\ 1000, \ 1000, 1000\ }; \ \ \ \ \ 
     (*\ Xt, \ Xc, \ Yt, \ Yc, \ S\ sao\ os\ valores\ de\ *) 

    Xc = {600, 600, 600}; \ \ \ \ \ \ \ \ \ 
     (*\ \ resistencia\ da\ lamina, \ nas\ direcoes\ 1, 2, *) 

    Yt = {\ 30, 30, \ 30}; \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ a\ tracao\ e\ compressao.\ S\ e'\ cisalhante\ *) 

    Yc = {\ 120, 120, \ 120}; \ 
S = {\ 70, 70, \ 70}; 

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

\( (*\ CALCULOS\ 

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

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

H

\(\( (*\ Inicializar\ areas\ *) 
q = Table[1, {i, 4}, {j, n}]; 
    \ \ \ \ \ \  (*\ Cada\ coluna\ contem\ matriz\ Q\ *) 

    qb = Table[1, {i, 6}, {j, n}]; \ \ \ \ \ 
     (*\ e\ Qbarra\ de\ cada\ lamina\ *) 

    ma = {0.0, 0.0, 0.0, 0.0, 0.0, 0.0}; \ 

    mbb = {0.0, 0.0, 0.0, 0.0, 0.0, 0.0}; 

    md = {0.0, 0.0, 0.0, 0.0, 0.0, 0.0}; 

    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ Matriz\ D.\ Valores\ em\ coluna\ da\ parte\ 
        triang.sup.
			\ \ \ \ \ \ \ \ \ \ \ \ \ d11\ d12\ d22\ d16\ d26\ 
        d66.\ O\ mesmo\ com\ qb.
			\ \ \ \ \ \ \ \ \ \ \ \ \ q11\ q12\ 
        q22\ q66\ \ \ \ \ \ \ \ *) \)\)

\( (*\ Calculo\ de\ q\ em\ cada\ lamina\ *) 

    Do[q[\([1, j]\)] = 
        e1[\([j]\)]\^2\/\(e1[\([j]\)] - nu[\([j]\)]\^2\ e2[\([j]\)]\); 
	
      q[\([2, j]\)] = 
        \(nu[\([j]\)]\ e1[\([j]\)]\ e2[\([j]\)]\)\/\(e1[\([j]\)] - 
            nu[\([j]\)]\^2\ e2[\([j]\)]\); 
	
      q[\([3, j]\)] = 
        \(\ e1[\([j]\)]\ e2[\([j]\)]\)\/\(e1[\([j]\)] - 
            nu[\([j]\)]\^2\ e2[\([j]\)]\); 
		
      q[\([4, j]\)] = g12[\([j]\)], {j, 1, n}]\)TableForm[q]

\(\( (*\ Calculo\ de\ qbarra\ em\ cada\ lamina\ *) 

    Do[c4 = Cos[teta[\([j]\)]\ ]^4; \ \ \ \ \ \ \ \ \ 
		\ \ 
      s4 = Sin[teta[\([j]\)]\ ]^4; 
	\ \ \ \ c2 = Cos[teta[\([j]\)]\ ]^2; 
      \ \ \ \ \ \ \ \ \ \ 
	\ 	\ s2 = Sin[teta[\([j]\)]\ ]^2; 
	\ \ \ \ 
      c3 = Cos[teta[\([j]\)]\ ]^3; \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ 	
      s3 = Sin[teta[\([j]\)]\ ]^3; 
	\ \ \ \ co = Cos[teta[\([j]\)]\ ]; 
      \ \ \ \ \ \ \ \ \ \ 
	\ 	\ se = Sin[teta[\([j]\)]\ ]; 
	
      qb[\([1, j]\)] = 
        q[\([1, j]\)]\ c4 + 2 \((q[\([2, j]\)] + 2  q[\([4, j]\)])\) s2\ c2\ 
           + q[\([3, j]\)] s4; 
	
      qb[\([3, j]\)] = \ 
        q[\([1, j]\)] s4 + 2 \((q[\([2, j]\)] + 2  q[\([4, j]\)])\) s2\ c2 + 
          q[\([3, j]\)] c4\ \ \ ; 
	
      qb[\([6, j]\)] = 
        \((q[\([1, j]\)] + q[\([3, j]\)] - 
                2 \((q[\([2, j]\)] + q[\([4, j]\)])\))\) s2\ c2 + 
          q[\([4, j]\)] \((s4 + c4)\)\ \ ; 
	
      qb[\([2, j]\)] = 
        \((q[\([1, j]\)] + q[\([3, j]\)] - 4  q[\([4, j]\)])\) s2\ c2 + 
          q[\([2, j]\)] \((s4 + c4)\); 
	
      qb[\([4, j]\)] = 
        \((q[\([1, j]\)] - q[\([2, j]\)] - 2  q[\([4, j]\)])\) se\ c3 + 
          \((q[\([2, j]\)] - q[\([3, j]\)] + 2  q[\([4, j]\)])\) s3\ co; 
	
      qb[\([5, j]\)] = 
        \((q[\([1, j]\)] - q[\([2, j]\)] - 2  q[\([4, j]\)])\) s3\ co + 
          \((q[\([2, j]\)] - q[\([3, j]\)] + 2  q[\([4, j]\)])\) se\ c3, 
      \ \ \ \ \ \ \ \ \ \ \ \ {j, 1, n}]\)\)

TableForm[q]\(TableForm[qb]
\)

\(\( (*\ CALCULO\ DAS\ MATRIZES\ DE\ RIGIDEZ\ A, \ B, \ e\ D\ *) 

    \(Do[
	Do[
		ma[\([i]\)] = ma[\([i]\)] + qb[\([i, j]\)]\ h[\([j]\)]; 
        
		mbb[\([i]\)] = 
          mbb[\([i]\)] + qb[\([i, j]\)]\ h[\([j]\)]\ z0[\([j]\)]; 
		
        md[\([i]\)] = 
          md[\([i]\)] + 
            qb[\([i, j]\)] 
              \((h[\([j]\)]\ z0[\([j]\)]\^2\  + h[\([j]\)]\^3/12.0)\)\ , 
        \ {j, 1, n}], {i, 1, 6}]; \)\)\)

TableForm[ma]TableForm[mbb]TableForm[md]Print["D22/ D11 ", md[[3]]/md[[1]], 
      "  (D12+2D66)/ D11 ", 
      (md[[2]] + 2  md[[6]])/md[[1]], "  D16/ D11 ",   
      md[[4]]/md[[1]] ]

\(\( (*\ 
      LAMINADO\ RETANGULAR\ SIMPLESMENTE\ APOIADO\ SOB\ CARGA\ UNIFORME\ 
        q0.\ TEORIA\ CLASSICA\ *) 
 (*\ inicializacoes\ *) 

    qmn = Table[0, {i, mh}, {j, nh}]; 
wmn = Table[0, {i, mh}, {j, nh}]; 
    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ coeficientes\ do\ deslocam.\ w\ *) 

    mk = Table[0, {i, mh\ nh}, {j, nh\ mh}]; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ matriz\ de\ rigidez\ *) 
for = Table[0, {i, mh\ nh}]; 
\)\)

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

     (*\ CARGA\ SENOIDAL\ qo\ \ USADO\ SO\ NO\ CASO\ mh\  = \ 
        \(nh\  = \ 1\)\ \ \ \ \ 

      \ \(Do[
\ \ \ \ 	
        Do[qmn[\([i, j]\)] = qo, {i, 1, mh, 2}], {j, 1, nh, 2}]; \)\ \ \ *) 

     (*\ TableForm[qmn]\ \ *) \)\)

\(\(\(Do[
      Do[Do[Do[\ \ ib = \(i\ Pi\)\/a; \ \ jb = \(j\ Pi\)\/b; \ \ \ \ \ \ \ \ 
            mb = \(mm\ Pi\)\/a; \ \ nb = \(nn\ Pi\)\/b; \ \ \ \ \ \ \ \ 

            				l = \((j - 1)\) mh + i; \ \ \ \ \ \ 
            k = \((nn - 1)\) nh + mm; 
				\ 
            If[i\  != \ mm\  && \ j\  != \ nn, 
					
              sim = \((
                    \(-2\)\ i + \((i + mm)\) \((\(-1\))\)^\((i - mm)\) + 
                      \((i - mm)\) \((\(-1\))\)^\((i + mm)\))\)\/\(2  Pi 
                    \((i - mm)\) \((i + mm)\)\); 
					
              sjn = \((
                    \(-2\)\ j + \((j + nn)\) \((\(-1\))\)^\((j - nn)\) + 
                      \((j - nn)\) \((\(-1\))\)^\((j + nn)\))\)\/\(2  Pi 
                    \((j - nn)\) \((j + nn)\)\); 
					
              cim = \((
                    \(-2\)\ mm + \((i + mm)\) \((\(-1\))\)^\((mm - i)\) + 
                      \((mm - i)\) \((\(-1\))\)^\((i + mm)\))\)\/\(2  Pi 
                    \((mm - i)\) \((i + mm)\)\); 
					
              cjn = \((
                    \(-2\)\ nn + \((j + nn)\) \((\(-1\))\)^\((nn - j)\) + 
                      \((nn - j)\) \((\(-1\))\)^\((j + nn)\))\)\/\(2  Pi 
                    \((nn - j)\) \((j + nn)\)\); 
					
              mk[\([l, k]\)] = 
                \(-2\)\ a\ b\ md[\([4]\)]\ 
                    \((mb\ nb\ ib^2\ sim\ sjn\  + \ ib\ jb\ mb^2\ cim\ cjn)
                      \) - 2\ a\ b\ md[\([5]\)] 
                    \((mb\ nb\ jb^2\ sim\ sjn + ib\ jb\ nb^2\ cim\ cjn)\)]; \ 
            
				
            If[i\  == \ mm\  && \ j == nn, 
					
              mk[\([l, k]\)] = 
                a\ b \((
                      md[\([1]\)]\ ib^4\  + 
                        2 \((md[\([2]\)] + 2  md[\([6]\)])\) \((ib\ jb)\)^2\ 
                         + \ \ md[\([3]\)] jb^4)\)/4]\ \ , {mm, 1, mh}], {nn, 
            1, nh}]; 
        for[\([l]\)] = \(\ qmn[\([i, j]\)] a\ b\ \)\/4, {i, 1, mh}], {j, 1, 
        nh}]; \)
 (*\ TableForm[for]

      TableForm[mk]\ \ \ *) \)\)

\(\(\ vsol = LinearSolve[mk, for\ ]; 
	
	
    Do[\ nn = Quotient[\((k - 1)\), mh] + 1; \ mm = k - \((nn - 1)\)\ mh; 
      \ 			wmn[\([mm, nn]\)]\ \ \ \ \ \ \  = vsol[\([k]\)], {k, 1, mh\ nh}]
      \)\)

\( (*\ 
      Deslocamento\ no\ ponto\ escolhido\ e\ o\ deslocamento\ maximo\ \ *) 

    wp = Sum[wmn[\([i, j]\)] Sin[\(i\ Pi\ xc\ \)\/a]\ 
          Sin[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 1, nh}]\)wpN = ((wp (H^3) e2[[1]] 100)/(qo (a^4)))wmax = 
      Sum[wmn[[i, j]] Sin[((i Pi )/2)] Sin[((j Pi )/2)], {i, 1, 
          mh}, {j, 1, nh}] wmaxN = ((wmax (H^3) e2[[1]] 100)/(qo (a^4)))  wmaxN = ((wmax   md[[1]])/(qo (a^4)))mk[[4, 4]]

\(\( (*\ \ TESTE : \ DEVERIA\ DAR\ 1\ \ \ *) 

     (*\ Sum[Sum[
            \(16\ \ \ Sin[\(i\ Pi\ \)\/2]\ Sin[\(j\ Pi\ \)\/2]\)\/\(Pi\ Pi\ i
                \ j\), {i, 1, 99, 2}], {j, 1, 99, 2}]\  // 
        N\ \ \(TESTE : \ DEVERIA\ DAR\ 1\)\ *) \)\)

\(\( (*\ 
      \(--\(--\(--
                \(--\(--
                    \(--\(--
                        \(--\(--
                            \(--\(--
                                \(--\(--
                                    \(--\(--
                                        \(--\(--
                                        \(--\(--
                                        \(--\(--\(--\(--
                                        \(--\(--\(--\(--\(--\(--\(--\(--
                                        \(--\(--\(--\(--\(--\(--\(--\(-

                                        													CALCULO
                                        \)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\ DOS\ DESLOCAMENTOS\ E\ ROTACOES\ NO
          \ PONTO\ \((x, y)\) = \((xc, yc)\)\ \ *) 

    wpx = Sum[
        wmn[\([i, j]\)] \((\(i\ Pi\ \)\/a)\) Cos[\(i\ Pi\ xc\ \)\/a]\ 
          Sin[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 1, nh}]; 

    wpxx = \(-
        Sum[wmn[\([i, j]\)] \(\((\(i\ Pi\ \)\/a)\)\^2\) 
            Sin[\(i\ Pi\ xc\ \)\/a]\ Sin[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 
            1, nh}]\); 

    wpxxx = \(-
        Sum[wmn[\([i, j]\)] \(\((\(i\ Pi\ \)\/a)\)\^3\) 
            Cos[\(i\ Pi\ xc\ \)\/a]\ Sin[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 
            1, nh}]\); 

    wpy = Sum[
        wmn[\([i, j]\)] \((\(j\ Pi\ \)\/b)\) Sin[\(i\ Pi\ xc\ \)\/a]\ 
          Cos[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 1, nh}]; 

    wpyy = \(-
        Sum[wmn[\([i, j]\)] \(\((\(j\ Pi\ \)\/b)\)\^2\) 
            Sin[\(i\ Pi\ xc\ \)\/a]\ Sin[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 
            1, nh}]\); 

    wpyyy = \(-
        Sum[wmn[\([i, j]\)] \(\((\(j\ Pi\ \)\/b)\)\^3\) 
            Sin[\(i\ Pi\ xc\ \)\/a]\ Cos[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 
            1, nh}]\); 

    wpxy = Sum[
        wmn[\([i, j]\)] \((\(i\ Pi\ \)\/a)\) \((\(j\ Pi\ \)\/b)\) 
          Cos[\(i\ Pi\ xc\ \)\/a] Cos[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 1, 
          nh}]; 
wpxxy = 
      \(-Sum[wmn[\([i, j]\)] \(\((\(i\ Pi\ \)\/a)\)\^2\) \((\(j\ Pi\ \)\/b)\) 
            Sin[\(i\ Pi\ xc\ \)\/a] Cos[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 
            1, nh}]\); 

    wpxyy = \(-
        Sum[wmn[\([i, j]\)] \((\(i\ Pi\ \)\/a)\) \(\((\(j\ Pi\ \)\/b)\)\^2\) 
            Cos[\(i\ Pi\ xc\ \)\/a] Sin[\(j\ Pi\ yc\ \)\/b], {i, 1, mh}, {j, 
            1, nh}]\); \)\)

\(psx = \(-wpx\); \ \ \ \ \ \ \ \ \ \ \ \ \ psy = \(-wpy\); \ \ \ \ \ \ 

    psxx = \(-wpxx\); \ \ \ \ \ \ \ \ psyy = \(-wpyy\); 
psxy = \(-wpxy\); 
    \ \ \ \ \ \ \ \ \ psyx = \(-wpxy\); 

psxxx = \(-wpxxx\); \ \ \ \ \ \ 
    psyyy = \(-wpyyy\); 
psxxy = \(-wpxxy\); \ \ \ \ \ \ \ 
    psyyx = \(-wpxyy\); 
psyxx = \(-wpxxy\); \ \ \ \ \ \ \ 
    psyxy = \(-wpxyy\); 
psxyy = \(-wpxyy\); 

 (*\ curvaturas\ *) 

    kxy\  = \ psxy + psyx; 
kxyx\  = \ psxxy + psyxx; 

    kxyy\  = \ psxyy + psyxy; \)

\(\( (*\ TENSOES\ *) 
sigx = Table[0, {i, 2  n}, {j, 2}]; \ \ 
     (*\ sigx\ na\ sup\ inf\ e\ sup\ de\ cada\ uma\ das\ n\ \(lam.\)*) 

    sigy = Table[0, {i, 2  n}, {j, 2}]; 

    sigxy = Table[0, {i, 2  n}, {j, 2}]; 

    tauxzi = Table[0, {i, 2  n}, {j, 2}]; \ \ 
     (*\ tensoes\ integradas\ \ *) 
tauyzi = Table[0, {i, 2  n}, {j, 2}]; 
    \)\)

\(\(Do[
      sigx[\([2  k - 1, 2]\)] = 
        z[\([k]\)] 
          \((qb[\([1, k]\)] psxx + qb[\([2, k]\)] psyy + 
              qb[\([4, k]\)] \((psxy + psyx)\))\); 
\ \ 
      sigx[\([2  k, 2]\)] = 
        z[\([k + 1]\)] 
          \((qb[\([1, k]\)] psxx + qb[\([2, k]\)] psyy + 
              qb[\([4, k]\)] \((psxy + psyx)\))\); 
	
\ 
      sigy[\([2  k - 1, 2]\)] = 
        z[\([k]\)] 
          \((qb[\([2, k]\)] psxx + qb[\([3, k]\)] psyy + 
              qb[\([5, k]\)] \((psxy + psyx)\))\); 
\ 
      sigy[\([2  k, 2]\)] = 
        z[\([k + 1]\)] 
          \((qb[\([2, k]\)] psxx + qb[\([3, k]\)] psyy + 
              qb[\([5, k]\)] \((psxy + psyx)\))\); 
	
\ 
      sigxy[\([2  k - 1, 2]\)] = 
        z[\([k]\)] 
          \((qb[\([4, k]\)] psxx + qb[\([5, k]\)] psyy + 
              qb[\([6, k]\)] \((psxy + psyx)\))\); 

      sigxy[\([2  k, 2]\)] = 
        z[\([k + 1]\)] 
          \((qb[\([4, k]\)] psxx + qb[\([5, k]\)] psyy + 
              qb[\([6, k]\)] \((psxy + psyx)\))\), 
      \ \ \ \ \ \ \ \ \ \ \ \ \ {k, n}]; \)\)

\(\( (*\ 
      Tensoes\ cisalhantes\ obtidas\ por\ integracao\ das\ eqs.equilibrio\ *) 
      

tauxzi[\([1, 2]\)] = 0; 
tauyzi[\([1, 2]\)] = 0; 

    Do[zk2 = z[\([k]\)]\^2/2; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
      zk12 = z[\([k + 1]\)]\^2/2; \ \ \ \ \ 
	
      tauxzi[\([2  k, 2]\)] = \ 
        tauxzi[\([2  k - 1, 2]\)] - 
	\ \ \ \ \ 
          \((\ \ qb[\([1, k]\)] zk12\ psxxx + qb[\([2, k]\)] zk12\ psyxy + \ 
              qb[\([4, k]\)]\ zk12\ kxyx + 
		\ \ \ \ 
              qb[\([4, k]\)] zk12\ psxxy + qb[\([5, k]\)] zk12\ psyyy + \ 
              qb[\([6, k]\)]\ zk12\ kxyy\ \ )\) + 
		\ \ 
          \((\ \ qb[\([1, k]\)] zk2\ psxxx + qb[\([2, k]\)] zk2\ psyxy + 
              qb[\([4, k]\)]\ zk2\ kxyx + 
		\ \ \ \ 
              qb[\([4, k]\)] zk2\ psxxy + qb[\([5, k]\)] zk2\ psyyy + 
              qb[\([6, k]\)]\ zk2\ kxyy\ \ )\); 
	
       (*\ superf.inf.da\ prox.lam.*) 
		
      If[k < n, tauxzi[\([2  k + 1, 2]\)] = tauxzi[\([2  k, 2]\)]]\ , {k, 
        n}]; \ 


    Do[zk2 = z[\([k]\)]\^2/2; \ \ \ \ \ \ \ \ \ \ \ \ \ 
      zk12 = z[\([k + 1]\)]\^2/2; \ \ 
	
      tauyzi[\([2  k, 2]\)] = \ 
        tauyzi[\([2  k - 1, 2]\)] - 
	\ \ \ \ \ 
          \((\ \ qb[\([2, k]\)] zk12\ psxxy + qb[\([3, k]\)] zk12\ psyyy + \ 
              qb[\([5, k]\)]\ zk12\ kxyy + 
		\ \ \ \ 
              qb[\([4, k]\)] zk12\ psxxx + qb[\([5, k]\)] zk12\ psyxy + 
              qb[\([6, k]\)]\ zk12\ kxyx\ \ )\) + 
		\ \ 
          \((\ \ qb[\([2, k]\)] zk2\ psxxy + qb[\([3, k]\)] zk2\ psyyy + 
              qb[\([5, k]\)]\ zk2\ kxyy + 
		\ \ \ \ 
              qb[\([4, k]\)] zk2\ psxxx + qb[\([5, k]\)] zk2\ psyxy + 
              qb[\([6, k]\)]\ zk2\ kxyx\ \ )\); 
		
      If[k < n, tauyzi[\([2  k + 1, 2]\)] = tauyzi[\([2  k, 2]\)]\ \ ]
        \ \ \ \ \ \ \ \ \ \ , {k, n}]; \)\)

\(sigxN = Table[0, {i, 2  n}, {j, 2}]; \ \ 
     (*\ sigx\ na\ sup\ inf\ e\ sup\ de\ cada\ uma\ das\ n\ \(lam.\)*) 

    sigyN = Table[0, {i, 2  n}, {j, 2}]; \ \ \ \ 
     (*\ valores\ normalizados\ *) 
sigxyN = Table[0, {i, 2  n}, {j, 2}]; 

    tauxziN = Table[0, {i, 2  n}, {j, 2}]; 

    tauyziN = Table[0, {i, 2  n}, {j, 2}]; 

    sigxN = sigx  H\^2\/\(qo\ a\^2\); 
sigyN = sigy  H\^2\/\(qo\ a\^2\); 

    sigxyN = sigxy  H\^2\/\(qo\ a\^2\); 
tauyziN = tauyzi\ H\/\(qo\ a\); 
    \ \ \ \ \ \ \ \ \ tauxziN = tauxzi\ H\/\(qo\ a\); 
		
    Do[
\ \ \ \ 	tauyzi[\([2  k - 1, 1]\)] = z[\([k]\)]; 
      \ \ \ \ \ \ \ \ \ \ tauyzi[\([2  k, 1]\)]\ \ \  = z[\([k + 1]\)]; 

      		\ \ tauxzi[\([2  k - 1, 1]\)] = z[\([k]\)]; \ \ \ \ \ \ \ \ \ 
      tauxzi[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 
		\ \ 
      tauyziN[\([2  k - 1, 1]\)] = z[\([k]\)]/H; \ 
      tauyziN[\([2  k, 1]\)]\  = z[\([k + 1]\)]/H; 
		\ \ 
      tauxziN[\([2  k - 1, 1]\)] = z[\([k]\)]/H; \ 
      tauxziN[\([2  k, 1]\)]\ \  = z[\([k + 1]\)]/H, 

      	\ \ \ \ \ \ \ \ {k, n}]; \)

\(\( (*\ TENSOES\ NAS\ DIRECOES\ PRINCIPAIS\ *) 

    sig1 = Table[0, {i, 2  n}, {j, 2}]; \ \ 
     (*\ sigx\ na\ sup\ inf\ e\ sup\ de\ cada\ uma\ das\ n\ \(lam.\)*) 

    sig2 = Table[0, {i, 2  n}, {j, 2}]; 

    sig12 = Table[0, {i, 2  n}, {j, 2}]; 

    Do[co = Cos[teta[\([k]\)]]; \ se = Sin[teta[\([k]\)]]; 
	\ \ \ 
      c2 = co\ co; s2 = se\ se; 
	\ \ 
      sig1[\([2  k - 1, 2]\)] = 
        c2\ sigx[\([2  k - 1, 2]\)] + s2\ sigy[\([2  k - 1, 2]\)] + 
          2\ se\ co\ sigxy[\([2  k - 1, 2]\)]; 
\ \ 	
      sig1[\([2  k, 2]\)] = 
        c2\ sigx[\([2  k, 2]\)] + s2\ sigy[\([2  k, 2]\)] + 
          2\ se\ co\ sigxy[\([2  k, 2]\)]; 
	
	
      sig2[\([2  k - 1, 2]\)] = 
        s2\ sigx[\([2  k - 1, 2]\)] + c2\ sigy[\([2  k - 1, 2]\)] - 
          2\ se\ co\ sigxy[\([2  k - 1, 2]\)]; 
	
      sig2[\([2  k, 2]\)] = 
        s2\ sigx[\([2  k, 2]\)] + c2\ sigy[\([2  k, 2]\)] - 
          2\ se\ co\ sigxy[\([2  k, 2]\)]; 
	
	
      sig12[\([2  k - 1, 2]\)] = 
        \(-se\)\ co\ sigx[\([2  k - 1, 2]\)] + 
          se\ co\ sigy[\([2  k - 1, 2]\)] + 
          \((c2 - s2)\)\ sigxy[\([2  k - 1, 2]\)]; 
		
      sig12[\([2  k, 2]\)] = 
        \(-se\)\ co\ sigx[\([2  k, 2]\)] + se\ co\ sigy[\([2  k, 2]\)] + 
          \((c2 - s2)\)\ sigxy[\([2  k, 2]\)], {k, n}]; \)\)

\(\(sig1N = Table[0, {i, 2  n}, {j, 2}]; \ \ 
     (*\ sig1\ na\ sup\ inf\ e\ sup\ de\ cada\ uma\ das\ n\ \(lam.\)*) 

    sig2N = Table[0, {i, 2  n}, {j, 2}]; \ \ \ \ \ \ 
     (*\ valores\ normalizados\ *) 
sig12N = Table[0, {i, 2  n}, {j, 2}]; 

    sig1N = sig1  H\^2\/\(qo\ a\^2\); 
sig2N = sig2  H\^2\/\(qo\ a\^2\); 

    sig12N = sig12  H\^2\/\(qo\ a\^2\); 
\)\)

\(\(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[\([k + 1]\)]; 
		\ \ 
      sigxy[\([2  k - 1, 1]\)] = z[\([k]\)]; 
	\ \ \ \ 
      sigxy[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 

      		\ \ \ \ \ \ \ \ \ \ \ sig1[\([2  k - 1, 1]\)] = z[\([k]\)]; 

      	\ \ \ \ sig1[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 
		\ \ 
      sig2[\([2  k - 1, 1]\)] = z[\([k]\)]; 
	\ \ \ \ 
      sig2[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 
		\ \ 
      sig12[\([2  k - 1, 1]\)] = z[\([k]\)]; 
	\ \ \ \ 
      sig12[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 

      		\ \ \ \ \ \ \ \ \ \ \ 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; 
		\ \ 
      sigxyN[\([2  k - 1, 1]\)] = z[\([k]\)]/H; 
	\ \ \ \ 
      sigxyN[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]/H; 

      		\ \ \ \ \ \ \ \ \ \ \ sig1N[\([2  k - 1, 1]\)] = z[\([k]\)]/H; 

      	\ \ \ \ sig1N[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]/H; 

      		\ \ sig2N[\([2  k - 1, 1]\)] = z[\([k]\)]/H; 
	\ \ \ \ 
      sig2N[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]/H; 
		\ \ 
      sig12N[\([2  k - 1, 1]\)] = z[\([k]\)]/H; 
	\ \ \ \ 
      sig12N[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]/H, 
      \ \ \ \ \ \ \ \ {k, n}]; \)\)

\(\( (*\ MOMENTOS\ FLETORES\ E\ TORCORES, \ Mx, \ My, \ Mxy\ \ *) 

    mx = md[\([1]\)] psxx + md[\([2]\)] psyy + md[\([4]\)] \((psxy + psyx)\); 
    
\ my = md[\([2]\)] psxx + md[\([3]\)] psyy + 
        md[\([5]\)] \((psxy + psyx)\); 
\ 
    mxy = md[\([4]\)] psxx + md[\([5]\)] psyy + 
        md[\([6]\)] \((psxy + psyx)\); 
		mxN = \(mx\ H\)\/\(qo\ a\^3\); \ 
     (*\ Valores\ normalizados\ *) 
	myN = \(mx\ H\)\/\(qo\ a\^3\); 
	
    mxyN = \(mxy\ H\)\/\(qo\ a\^3\); \)\)

\( (*\ FATORES\ DE\ SOLICITACAO\ *) 


    ftm = Table[0, {i, 2  n}, {j, 2}]; \ \ 
     (*\ fatores\ em\ cada\ interface\ para\ tensao\ maxima\ *) 

    fth = Table[0, {i, 2  n}, {j, 2}]; \ \ \ \ \ 
     (*\ fator\ para\ Tsai - Hill\ \ *) 
ftw = Table[0, {i, 2  n}, {j, 2}]; 
    \ \ \ \ \  (*\ fator\ para\ Tsai - Wu\ \ *) 

    Do[
	 (*\ CRITERIO\ DA\ TENSAO\ MAXIMA\ *) 

      ftm[\([2  k - 1, 2]\)] = 
        Max[{If[\ sig1[\([2  k - 1, 2]\)] >= 0, 
              sig1[\([2  k - 1, 2]\)]\/Xt[\([k]\)], \ 
              \(-sig1[\([2  k - 1, 2]\)]\)\/Xc[\([k]\)]], 
            If[\ sig2[\([2  k - 1, 2]\)] >= 0, 
              sig2[\([2  k - 1, 2]\)]\/Yt[\([k]\)], \ 
              \(-sig2[\([2  k - 1, 2]\)]\)\/Yc[\([k]\)]]\ , 
            Abs[sig12[\([2  k - 1, 2]\)]]\/S[\([k]\)]}]; 
		
      ftm[\([2  k, 2]\)] = 
        Max[{If[\ sig1[\([2  k, 2]\)] >= 0, sig1[\([2  k, 2]\)]\/Xt[\([k]\)], 
              \ \(-sig1[\([2  k, 2]\)]\)\/Xc[\([k]\)]], 
            If[\ sig2[\([2  k, 2]\)] >= 0, sig2[\([2  k, 2]\)]\/Yt[\([k]\)], 
              \ \(-sig2[\([2  k, 2]\)]\)\/Yc[\([k]\)]]\ , 
            Abs[sig12[\([2  k, 2]\)]]\/S[\([k]\)]}]; 
	
       (*\ CRITERIO\ DE\ TSAI - HILL\ *) 
	
	
      fth[\([2  k, 2]\)] = 	
        If[\ sig1[\([2  k, 2]\)] >= 0, sig1[\([2  k, 2]\)]\/Xt[\([k]\)], \ 
              \(-sig1[\([2  k, 2]\)]\)\/Xc[\([k]\)]]^2 + 
			
          If[\ sig2[\([2  k, 2]\)] >= 0, sig2[\([2  k, 2]\)]\/Yt[\([k]\)], \ 
              \(-sig2[\([2  k, 2]\)]\)\/Yc[\([k]\)]]^2 + 
			
          \((sig12[\([2  k, 2]\)]\/S[\([k]\)])\)^2 - 
			
          If[\ sig1[\([2  k, 2]\)] >= 0, sig1[\([2  k, 2]\)]\/Xt[\([k]\)], \ 
              \(-sig1[\([2  k, 2]\)]\)\/Xc[\([k]\)]] 
            If[\ sig2[\([2  k, 2]\)] >= 0, sig2[\([2  k, 2]\)]\/Yt[\([k]\)], 
              \ \(-sig2[\([2  k, 2]\)]\)\/Yc[\([k]\)]]; 
	
		
      fth[\([2  k - 1, 2]\)] = 	
        If[\ sig1[\([2  k - 1, 2]\)] >= 0, 
              sig1[\([2  k - 1, 2]\)]\/Xt[\([k]\)], \ 
              \(-sig1[\([2  k - 1, 2]\)]\)\/Xc[\([k]\)]]^2 + 
			
          If[\ sig2[\([2  k - 1, 2]\)] >= 0, 
              sig2[\([2  k - 1, 2]\)]\/Yt[\([k]\)], \ 
              \(-sig2[\([2  k - 1, 2]\)]\)\/Yc[\([k]\)]]^2 + 
			
          \((sig12[\([2  k - 1, 2]\)]\/S[\([k]\)])\)^2 - 
			
          If[\ sig1[\([2  k - 1, 2]\)] >= 0, 
              sig1[\([2  k - 1, 2]\)]\/Xt[\([k]\)], \ 
              \(-sig1[\([2  k - 1, 2]\)]\)\/Xc[\([k]\)]] 
            If[\ sig2[\([2  k - 1, 2]\)] >= 0, 
              sig2[\([2  k - 1, 2]\)]\/Yt[\([k]\)], \ 
              \(-sig2[\([2  k - 1, 2]\)]\)\/Yc[\([k]\)]]; 
		
       (*\ CRITERIO\ TSAI - WU\ *) 

      ftw[\([2  k - 1, 2]\)] = 
        \((1\/Xt[\([k]\)] - 1\/Xc[\([k]\)])\) sig1[\([2  k - 1, 2]\)] + 
          \((1\/Yt[\([k]\)] - 1\/Yc[\([k]\)])\) sig2[\([2  k - 1, 2]\)] + 

          			
	sig1[\([2  k - 1, 2]\)]^2\/\(Xt[\([k]\)] Xc[\([k]\)]\) + 
          sig2[\([2  k - 1, 2]\)]^2\/\(Yt[\([k]\)] Yc[\([k]\)]\) + 
          \((sig12[\([2  k - 1, 2]\)]\/S[\([k]\)])\)^2 - 
			
          \(sig1[\([2  k - 1, 2]\)] 
              sig2[\([2  k - 1, 2]\)]\)\/\@\(Xt[\([k]\)] Xc[\([k]\)] 
                Yt[\([k]\)] Yc[\([k]\)]\); 
	
      ftw[\([2  k, 2]\)] = 
        \((1\/Xt[\([k]\)] - 1\/Xc[\([k]\)])\) sig1[\([2  k, 2]\)] + 
          \((1\/Yt[\([k]\)] - 1\/Yc[\([k]\)])\) sig2[\([2  k, 2]\)] + 
			
          
	sig1[\([2  k, 2]\)]^2\/\(Xt[\([k]\)] Xc[\([k]\)]\) + 
          sig2[\([2  k, 2]\)]^2\/\(Yt[\([k]\)] Yc[\([k]\)]\) + 
          \((sig12[\([2  k, 2]\)]\/S[\([k]\)])\)^2 - 
			
          \(sig1[\([2  k, 2]\)] 
              sig2[\([2  k, 2]\)]\)\/\@\(Xt[\([k]\)] Xc[\([k]\)] Yt[\([k]\)] 
                Yc[\([k]\)]\); 
	
	\ \ 
      fth[\([2  k - 1, 2]\)] = \@fth[\([2  k - 1, 2]\)]; 
		
      fth[\([2  k, 2]\)] = \@fth[\([2  k, 2]\)]; 
	\ 
      ftw[\([2  k - 1, 2]\)] = \@Abs[ftw[\([2  k - 1, 2]\)]]; 
		
      ftw[\([2  k, 2]\)] = \@Abs[ftw[\([2  k, 2]\)]]
	, 
			{k, n}]; 

    \ TableForm[ftm]\)\(TableForm[fth]
 (*\ TableForm[ftw]*) \)

\(\( (*\ Coeficiente\ de\ seguranca\ *) 
maxtm = Max[ftm]; 

    maxth = Max[fth]; 
If[maxtm == 0, cstm = 0, cstm = 1/maxtm]; 

    If[maxth == 0, csth = 0, csth = 1/maxth]; \)\)

\(\(Do[ftm[\([2  k - 1, 1]\)] = z[\([k]\)]; 
	\ \ \ \ 
      ftm[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)]; 
	
      fth[\([2  k - 1, 1]\)] = z[\([k]\)]; 
	\ \ \ \ 
      fth[\([2  k, 1]\)]\ \ \ \ \ \ \  = z[\([k + 1]\)], {k, n}]; \)\)

wmax
                                                  
     (* valor maximo do deslocamento,  no centro. *) wmaxNwp                                                   
     (* deslocamento no ponto (xc, yc) dado  *) wpNPrint["coef.seg tensao maxima ", cstm]Print["coef.seg Tsai-Hill     ", csth]

Print[" Momentos: Mx  = ", mx,   ", My =  ", my,   
      ", Mxy =  ", mxy]Print[" Momentos: MxN = ", mxN, ", MyN = ", myN, 
      ", MxyN = ", mxyN]

TableForm[sig1N]                     
      TableForm[sig2N]                   
      TableForm[sig1]

ListPlot[sigxN,PlotJoined -> True,GridLines -> Automatic,Frame -> True,AxesLabel -> {"sigxN", "zN"}]ListPlot[sig1N,PlotJoined -> True,GridLines -> Automatic,Frame -> True,AxesLabel -> {"sig1N", "zN"}]

TableForm[sigxN]                     
      \(TableForm[sigyN]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
     (*\ tensoes\ \ na\ direcao\ \ y\ *) 
\)

TableForm[fth]	ListPlot[fth,PlotJoined -> True,GridLines -> Automatic,Frame -> True,AxesLabel -> {"Fator Tsai-Hill.", "z"}]

TableForm[tauyziN]ListPlot[tauyziN,PlotJoined -> True]

mhnh