07_MEF_Elemento_Rectangular_Lineal.ipynb

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   "source": [
    " # Elemento rectangular de 4 nodos en elasticidad plana\n",
    "\n",
    "Análisis de Estructuras. Grado en Ingeniería Civil. \n",
    "\n",
    "Autores: \n",
    "\n",
    "+ Alejandro E. Martínez-Castro, Esther Puertas García, Rafael Gallego.\n",
    "+ Departamento de Mecánica de Estructuras e Ingeniería Hidráulica. Universidad de Granada.\n",
    "\n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\"><img alt=\"Licencia de Creative Commons\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">Cuadernos MEF: Elemento rectangular de 4 nodos</span> by <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">Alejandro Martínez Castro, Esther Puertas García, Rafael Gallego Sevilla</span> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional License</a>.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
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   "source": [
    "## Interpolación de desplazamientos y deformaciones. \n",
    "\n",
    "El campo de desplazamiento se determinará como\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{u} = \\mathbf{N}\\cdot \\mathbf{a}\n",
    "\\end{equation}\n",
    "\n",
    "En este elemento, los desplazamientos se interpolan a partir de los valores de $u$ y $v$ en cuatro nodos:\n",
    "\n",
    "Consideremos un elemento de 4 nodos. El elemento es rectangular, de base $2a$ y altura $2b$. \n",
    "La interpolación se establece a partir de: \n",
    "\n",
    "- Las coordenadas de los 4 vértices, $(-a, -b),(a,-b),(a,b),(-a,b)$. \n",
    "- Los desplazamientos en los 4 nodos, $(u_1,v_1),(u_2,v_2),(u_3,v_3),(u_4,v_4)$. \n",
    "\n",
    "Para este elemento, es posible trabajar en un sistema de coordenadas $(x,y)$ en un tratamiento local. \n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{pmatrix} u \\\\ v \\end{pmatrix} = \n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "N_1 & 0 & N_2 & 0 & N_3 & 0 & N_4 & 0 \\\\\n",
    "0 & N_1 & 0 & N_2 & 0 & N_3 & 0 & N_4 \n",
    "\\end{matrix}\n",
    "\\right] \\cdot \\begin{pmatrix} u_1 \\\\ v_1 \\\\ u_2 \\\\ v_2 \\\\ u_3 \\\\ v_3 \\\\ u_4 \\\\ v_4 \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Esta relación inducirá una relación para el vector $\\boldsymbol{\\varepsilon}$, \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\varepsilon} = \\mathbf{B}\\cdot \\mathbf{a}\n",
    "\\end{equation}"
   ]
  },
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   "source": [
    "## Funciones de forma\n",
    "\n",
    "A continuación se obtiene las funciones de forma. \n",
    "\n",
    "Para este elemento, la base $\\mathcal{B} = \\lbrace 1,x,y,x\\,y \\rbrace$, es decir $[1,x]\\times[1,y]$\n",
    "\n",
    "Esta es una base completa de los polinomios de grado parcial 1 sobre $(x,y)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0 (Python 2.7.13-32-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.0/\n"
     ]
    }
   ],
   "source": [
    "from sympy import init_session\n",
    "init_session()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "A continuación se determinarán las funciones de forma, siguiendo la metodología general que se ha explicado en cuadernos anteriores. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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      "text/latex": [
       "$$\\left[\\begin{matrix}1 & - a & - b & a b\\\\1 & a & - b & - a b\\\\1 & a & b & a b\\\\1 & - a & b & - a b\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  -a  -b  a⋅b ⎤\n",
       "⎢               ⎥\n",
       "⎢1  a   -b  -a⋅b⎥\n",
       "⎢               ⎥\n",
       "⎢1  a   b   a⋅b ⎥\n",
       "⎢               ⎥\n",
       "⎣1  -a  b   -a⋅b⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "base = Matrix([[1,x,y,x*y]])\n",
    "a,b = symbols('a b')\n",
    "matM = Matrix.vstack(base.subs([(x,-a),(y,-b)]),\n",
    "                     base.subs([(x,a),(y,-b)]),\n",
    "                     base.subs([(x,a),(y,b)]),\n",
    "                     base.subs([(x,-a),(y,b)]))\n",
    "matM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "outputs": [
    {
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      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4} & \\frac{1}{4}\\\\- \\frac{1}{4 a} & \\frac{1}{4 a} & \\frac{1}{4 a} & - \\frac{1}{4 a}\\\\- \\frac{1}{4 b} & - \\frac{1}{4 b} & \\frac{1}{4 b} & \\frac{1}{4 b}\\\\\\frac{1}{4 a b} & - \\frac{1}{4 a b} & \\frac{1}{4 a b} & - \\frac{1}{4 a b}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 1/4    1/4    1/4    1/4 ⎤\n",
       "⎢                          ⎥\n",
       "⎢ -1      1      1     -1  ⎥\n",
       "⎢ ───    ───    ───    ─── ⎥\n",
       "⎢ 4⋅a    4⋅a    4⋅a    4⋅a ⎥\n",
       "⎢                          ⎥\n",
       "⎢ -1     -1      1      1  ⎥\n",
       "⎢ ───    ───    ───    ─── ⎥\n",
       "⎢ 4⋅b    4⋅b    4⋅b    4⋅b ⎥\n",
       "⎢                          ⎥\n",
       "⎢  1     -1      1     -1  ⎥\n",
       "⎢─────  ─────  ─────  ─────⎥\n",
       "⎣4⋅a⋅b  4⋅a⋅b  4⋅a⋅b  4⋅a⋅b⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Inv_matM = matM.inv()\n",
    "Inv_matM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "N1 = base.dot(Inv_matM.col(0))\n",
    "N2 = base.dot(Inv_matM.col(1))\n",
    "N3 = base.dot(Inv_matM.col(2))\n",
    "N4 = base.dot(Inv_matM.col(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "A continuación se muestran las expresiones de las funciones de forma. Nótese que estas expresiones pueden utilizarse ya directamente, y se han deducido en las líneas anteriores."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\frac{1}{4} - \\frac{y}{4 b} - \\frac{x}{4 a} + \\frac{x y}{4 a b}$$"
      ],
      "text/plain": [
       "1    y     x     x⋅y \n",
       "─ - ─── - ─── + ─────\n",
       "4   4⋅b   4⋅a   4⋅a⋅b"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK0AAAArBAMAAAD4aBW1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/AYzLfMRq9TkRwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{4} - \\frac{y}{4 b} + \\frac{x}{4 a} - \\frac{x y}{4 a b}$$"
      ],
      "text/plain": [
       "1    y     x     x⋅y \n",
       "─ - ─── + ─── - ─────\n",
       "4   4⋅b   4⋅a   4⋅a⋅b"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK0AAAArBAMAAAD4aBW1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{1}{4} + \\frac{y}{4 b} + \\frac{x}{4 a} + \\frac{x y}{4 a b}$$"
      ],
      "text/plain": [
       "1    y     x     x⋅y \n",
       "─ + ─── + ─── + ─────\n",
       "4   4⋅b   4⋅a   4⋅a⋅b"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK0AAAArBAMAAAD4aBW1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAE1A8xEWFGXfAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{4} + \\frac{y}{4 b} - \\frac{x}{4 a} - \\frac{x y}{4 a b}$$"
      ],
      "text/plain": [
       "1    y     x     x⋅y \n",
       "─ + ─── - ─── - ─────\n",
       "4   4⋅b   4⋅a   4⋅a⋅b"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "IMPORTANTE: En las anteriores expresiones, nótese que los lados del elemento no son \"a\" y \"b\", sino \"2a\", \"2b\". Es importante tener esto en cuenta cuando se quiera aplicar. \n",
    "\n",
    "Para transcribir estas expresiones directamente en código Python, puede hacerse mediante el comando \"print\". "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1/4 - y/(4*b) - x/(4*a) + x*y/(4*a*b)\n",
      "1/4 - y/(4*b) + x/(4*a) - x*y/(4*a*b)\n",
      "1/4 + y/(4*b) + x/(4*a) + x*y/(4*a*b)\n",
      "1/4 + y/(4*b) - x/(4*a) - x*y/(4*a*b)\n"
     ]
    }
   ],
   "source": [
    "print N1\n",
    "print N2\n",
    "print N3\n",
    "print N4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Nótese que en las expresiones anteriores aparece la fracción 1/4. Para que se evalúe como 0.25 en versiones de Python 2 es necesario incluir al inicio de los códigos la expresión: \n",
    "\n",
    "        from __future__ import division"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Vector de deformaciones\n",
    "\n",
    "El vector de deformaciones se expresa mediante el siguiente producto: \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\varepsilon} = \\begin{pmatrix} \\varepsilon_{nx} \\\\ \\varepsilon_{ny} \\\\ \\gamma_{xy} \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Y en función de las funciones de forma: \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\varepsilon} = \\left[ \n",
    "\\begin{matrix}\n",
    "\\dfrac{\\partial N_1}{\\partial x} & 0 & \\dfrac{\\partial N_2}{\\partial x} & 0 & \\dfrac{\\partial N_3}{\\partial x} & 0 & \\dfrac{\\partial N_4}{\\partial x} & 0 \\\\\n",
    "0 & \\dfrac{\\partial N_1}{\\partial y} & 0 & \\dfrac{\\partial N_2}{\\partial y} & 0 & \\dfrac{\\partial N_3}{\\partial y} & 0 & \\dfrac{\\partial N_4}{\\partial y} \\\\\n",
    "\\dfrac{\\partial N_1}{\\partial y} & \\dfrac{\\partial N_1}{\\partial x} & \\dfrac{\\partial N_2}{\\partial y} & \\dfrac{\\partial N_2}{\\partial x} & \\dfrac{\\partial N_3}{\\partial y} & \\dfrac{\\partial N_3}{\\partial x} & \\dfrac{\\partial N_4}{\\partial y} & \\dfrac{\\partial N_4}{\\partial x}\n",
    "\\end{matrix} \\right] \\cdot\n",
    "\\begin{pmatrix} u_1 \\\\ v_1 \\\\ u_2 \\\\ v_2 \\\\ u_3 \\\\ v_3 \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Relación que puede escribirse como: $\\boldsymbol{\\varepsilon} = \\mathbf{B} \\cdot \\mathbf{a}$.\n",
    "\n",
    "La matriz $\\mathbf{B}$ en este contexto se denomina _matriz de deformación_.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{1}{4 a} + \\frac{y}{4 a b} & 0 & \\frac{1}{4 a} - \\frac{y}{4 a b} & 0 & \\frac{1}{4 a} + \\frac{y}{4 a b} & 0 & - \\frac{1}{4 a} - \\frac{y}{4 a b} & 0\\\\0 & - \\frac{1}{4 b} + \\frac{x}{4 a b} & 0 & - \\frac{1}{4 b} - \\frac{x}{4 a b} & 0 & \\frac{1}{4 b} + \\frac{x}{4 a b} & 0 & \\frac{1}{4 b} - \\frac{x}{4 a b}\\\\- \\frac{1}{4 b} + \\frac{x}{4 a b} & - \\frac{1}{4 a} + \\frac{y}{4 a b} & - \\frac{1}{4 b} - \\frac{x}{4 a b} & \\frac{1}{4 a} - \\frac{y}{4 a b} & \\frac{1}{4 b} + \\frac{x}{4 a b} & \\frac{1}{4 a} + \\frac{y}{4 a b} & \\frac{1}{4 b} - \\frac{x}{4 a b} & - \\frac{1}{4 a} - \\frac{y}{4 a b}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   1      y                     1      y                     1      y        \n",
       "⎢- ─── + ─────        0         ─── - ─────         0        ─── + ─────      \n",
       "⎢  4⋅a   4⋅a⋅b                  4⋅a   4⋅a⋅b                  4⋅a   4⋅a⋅b      \n",
       "⎢                                                                             \n",
       "⎢                  1      x                      1      x                  1  \n",
       "⎢      0        - ─── + ─────        0        - ─── - ─────       0       ─── \n",
       "⎢                 4⋅b   4⋅a⋅b                   4⋅b   4⋅a⋅b               4⋅b \n",
       "⎢                                                                             \n",
       "⎢   1      x       1      y       1      x      1      y      1      x     1  \n",
       "⎢- ─── + ─────  - ─── + ─────  - ─── - ─────   ─── - ─────   ─── + ─────  ─── \n",
       "⎣  4⋅b   4⋅a⋅b    4⋅a   4⋅a⋅b    4⋅b   4⋅a⋅b   4⋅a   4⋅a⋅b   4⋅b   4⋅a⋅b  4⋅a \n",
       "\n",
       "            1      y                 ⎤\n",
       " 0       - ─── - ─────        0      ⎥\n",
       "           4⋅a   4⋅a⋅b               ⎥\n",
       "                                     ⎥\n",
       "    x                     1      x   ⎥\n",
       "+ ─────        0         ─── - ───── ⎥\n",
       "  4⋅a⋅b                  4⋅b   4⋅a⋅b ⎥\n",
       "                                     ⎥\n",
       "    y      1      x        1      y  ⎥\n",
       "+ ─────   ─── - ─────   - ─── - ─────⎥\n",
       "  4⋅a⋅b   4⋅b   4⋅a⋅b     4⋅a   4⋅a⋅b⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nu = Matrix([[N1, 0, N2, 0, N3, 0, N4, 0]])\n",
    "Nv = Matrix([[0, N1, 0, N2, 0, N3, 0, N4]])\n",
    "B = Matrix.vstack(Nu.diff(x),Nv.diff(y),Nu.diff(y)+Nv.diff(x))\n",
    "B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "**Observación**: En este elemento, la matriz $\\mathbf{B}$ no es constante. Observe la dependencia de las variables $x$ e $y$.\n",
    "\n",
    "El elemento de 4 nodos rectangular genera una deformación no constante en el elemento. \n",
    "\n",
    "La variación es lineal. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Matriz de rigidez\n",
    "\n",
    "La expresión de la matriz de rigidez en un elemento es\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K} = \\int_{\\Omega} \\mathbf{B}^T \\cdot \\mathbf{D} \\cdot \\mathbf{B} \\, d \\Omega \n",
    "\\end{equation}\n",
    "\n",
    "Siendo $\\mathbf{D}$ la matriz de constantes elásticas de elasticidad plana. \n",
    "\n",
    "Puesto que el elemento es rectangular, esta integral puede expresarse como: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K} = \\int_{x=-a}^{x=a} \\int_{y=-b}^{y=b} \\mathbf{B}^T \\cdot \\mathbf{D} \\cdot \\mathbf{B} \\, dx \\,dy \n",
    "\\end{equation}\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "d11,d12,d33 = symbols('d11 d12 d33')\n",
    "D = Matrix([[d11, d12, 0],[d12,d11,0],[0,0,d33]])\n",
    "\n",
    "krigidez = integrate(B.T.multiply(D).multiply(B),(x,-a,a)) # Primero se integra en x, y luego en y\n",
    "krigidez = integrate(krigidez,(y,-b,b))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La matriz de rigidez $\\mathbf{K}$ puede expresarse como: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K} = \\left[ \\begin{matrix}\n",
    "\\mathbf{K_{11}} & \\mathbf{K_{12}} & \\mathbf{K_{13}} & \\mathbf{K_{14}}\\\\\n",
    "\\mathbf{K_{21}} & \\mathbf{K_{22}} & \\mathbf{K_{23}} & \\mathbf{K_{24}}\\\\\n",
    "\\mathbf{K_{31}} & \\mathbf{K_{32}} & \\mathbf{K_{33}} & \\mathbf{K_{34}} \\\\\n",
    "\\mathbf{K_{41}} & \\mathbf{K_{42}} & \\mathbf{K_{43}} & \\mathbf{K_{44}}\n",
    "\\end{matrix} \\right] \n",
    "\\end{equation}\n",
    "\n",
    "Con \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_{ij}} = \\int_{x=-a}^{x=a} \\int_{y=-b}^{y=b} \\mathbf{B_i}^T \\cdot \\mathbf{D} \\cdot \\mathbf{B_j} \\,dx\\,dy\n",
    "\\end{equation}\n",
    "\n",
    "Con la relación: \n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_{ji}} = \\mathbf{K_{ij}}^T\n",
    "\\end{equation}\n",
    "\n",
    "Esta matriz es simétrica. En las siguientes líneas se proporcionan las expresiones de los términos de la matriz triangular superior, pues son los únicos necesarios para definir completamente esta matriz. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "En lo que sigue se transcribirán las expresiones de las submatrices de rigidez de la triangular superior. Esto permite construir la matriz o componentes cualesquiera."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "k11 = krigidez[0:2,0:2]\n",
    "k22 = krigidez[2:4,2:4]\n",
    "k33 = krigidez[4:6,4:6]\n",
    "k44 = krigidez[6:8,6:8]\n",
    "\n",
    "k12 = krigidez[0:2,2:4]\n",
    "k13 = krigidez[0:2,4:6]\n",
    "k14 = krigidez[0:2,6:8]\n",
    "\n",
    "k23 = krigidez[2:4,4:6]\n",
    "k24 = krigidez[2:4,6:8]\n",
    "\n",
    "k34 = krigidez[4:6,6:8]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOcAAAA1CAMAAACN3Aw2AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{a d_{33}}{3 b} + \\frac{b d_{11}}{3 a} & \\frac{d_{12}}{4} + \\frac{d_{33}}{4}\\\\\\frac{d_{12}}{4} + \\frac{d_{33}}{4} & \\frac{a d_{11}}{3 b} + \\frac{b d_{33}}{3 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅d₃₃   b⋅d₁₁    d₁₂   d₃₃  ⎤\n",
       "⎢───── + ─────    ─── + ───  ⎥\n",
       "⎢ 3⋅b     3⋅a      4     4   ⎥\n",
       "⎢                            ⎥\n",
       "⎢  d₁₂   d₃₃    a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢  ─── + ───    ───── + ─────⎥\n",
       "⎣   4     4      3⋅b     3⋅a ⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k11.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOcAAAA1CAMAAACN3Aw2AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\ne0GsP+E/fChBn5HXgvsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{a d_{33}}{3 b} + \\frac{b d_{11}}{3 a} & - \\frac{d_{12}}{4} - \\frac{d_{33}}{4}\\\\- \\frac{d_{12}}{4} - \\frac{d_{33}}{4} & \\frac{a d_{11}}{3 b} + \\frac{b d_{33}}{3 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅d₃₃   b⋅d₁₁     d₁₂   d₃₃ ⎤\n",
       "⎢───── + ─────   - ─── - ─── ⎥\n",
       "⎢ 3⋅b     3⋅a       4     4  ⎥\n",
       "⎢                            ⎥\n",
       "⎢   d₁₂   d₃₃   a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢ - ─── - ───   ───── + ─────⎥\n",
       "⎣    4     4     3⋅b     3⋅a ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k22.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOcAAAA1CAMAAACN3Aw2AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{a d_{33}}{3 b} + \\frac{b d_{11}}{3 a} & \\frac{d_{12}}{4} + \\frac{d_{33}}{4}\\\\\\frac{d_{12}}{4} + \\frac{d_{33}}{4} & \\frac{a d_{11}}{3 b} + \\frac{b d_{33}}{3 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅d₃₃   b⋅d₁₁    d₁₂   d₃₃  ⎤\n",
       "⎢───── + ─────    ─── + ───  ⎥\n",
       "⎢ 3⋅b     3⋅a      4     4   ⎥\n",
       "⎢                            ⎥\n",
       "⎢  d₁₂   d₃₃    a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢  ─── + ───    ───── + ─────⎥\n",
       "⎣   4     4      3⋅b     3⋅a ⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k33.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOcAAAA1CAMAAACN3Aw2AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\ne0GsP+E/fChBn5HXgvsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{a d_{33}}{3 b} + \\frac{b d_{11}}{3 a} & - \\frac{d_{12}}{4} - \\frac{d_{33}}{4}\\\\- \\frac{d_{12}}{4} - \\frac{d_{33}}{4} & \\frac{a d_{11}}{3 b} + \\frac{b d_{33}}{3 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅d₃₃   b⋅d₁₁     d₁₂   d₃₃ ⎤\n",
       "⎢───── + ─────   - ─── - ─── ⎥\n",
       "⎢ 3⋅b     3⋅a       4     4  ⎥\n",
       "⎢                            ⎥\n",
       "⎢   d₁₂   d₃₃   a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢ - ─── - ───   ───── + ─────⎥\n",
       "⎣    4     4     3⋅b     3⋅a ⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k44.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOcAAAA1CAMAAACN3Aw2AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{a d_{33}}{6 b} - \\frac{b d_{11}}{3 a} & \\frac{d_{12}}{4} - \\frac{d_{33}}{4}\\\\- \\frac{d_{12}}{4} + \\frac{d_{33}}{4} & \\frac{a d_{11}}{6 b} - \\frac{b d_{33}}{3 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅d₃₃   b⋅d₁₁    d₁₂   d₃₃  ⎤\n",
       "⎢───── - ─────    ─── - ───  ⎥\n",
       "⎢ 6⋅b     3⋅a      4     4   ⎥\n",
       "⎢                            ⎥\n",
       "⎢   d₁₂   d₃₃   a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢ - ─── + ───   ───── - ─────⎥\n",
       "⎣    4     4     6⋅b     3⋅a ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k12.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQcAAAA1CAMAAAB7sN6HAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{a d_{33}}{6 b} - \\frac{b d_{11}}{6 a} & - \\frac{d_{12}}{4} - \\frac{d_{33}}{4}\\\\- \\frac{d_{12}}{4} - \\frac{d_{33}}{4} & - \\frac{a d_{11}}{6 b} - \\frac{b d_{33}}{6 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  a⋅d₃₃   b⋅d₁₁      d₁₂   d₃₃  ⎤\n",
       "⎢- ───── - ─────    - ─── - ───  ⎥\n",
       "⎢   6⋅b     6⋅a        4     4   ⎥\n",
       "⎢                                ⎥\n",
       "⎢    d₁₂   d₃₃      a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢  - ─── - ───    - ───── - ─────⎥\n",
       "⎣     4     4        6⋅b     6⋅a ⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k13.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQcAAAA1CAMAAAB7sN6HAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{a d_{33}}{3 b} + \\frac{b d_{11}}{6 a} & - \\frac{d_{12}}{4} + \\frac{d_{33}}{4}\\\\\\frac{d_{12}}{4} - \\frac{d_{33}}{4} & - \\frac{a d_{11}}{3 b} + \\frac{b d_{33}}{6 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  a⋅d₃₃   b⋅d₁₁      d₁₂   d₃₃  ⎤\n",
       "⎢- ───── + ─────    - ─── + ───  ⎥\n",
       "⎢   3⋅b     6⋅a        4     4   ⎥\n",
       "⎢                                ⎥\n",
       "⎢   d₁₂   d₃₃       a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢   ─── - ───     - ───── + ─────⎥\n",
       "⎣    4     4         3⋅b     6⋅a ⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k14.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQcAAAA1CAMAAAB7sN6HAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nMJbIoasBFw26GvqSdbXfTlxG3EJAIJkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{a d_{33}}{3 b} + \\frac{b d_{11}}{6 a} & \\frac{d_{12}}{4} - \\frac{d_{33}}{4}\\\\- \\frac{d_{12}}{4} + \\frac{d_{33}}{4} & - \\frac{a d_{11}}{3 b} + \\frac{b d_{33}}{6 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  a⋅d₃₃   b⋅d₁₁     d₁₂   d₃₃   ⎤\n",
       "⎢- ───── + ─────     ─── - ───   ⎥\n",
       "⎢   3⋅b     6⋅a       4     4    ⎥\n",
       "⎢                                ⎥\n",
       "⎢    d₁₂   d₃₃      a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢  - ─── + ───    - ───── + ─────⎥\n",
       "⎣     4     4        3⋅b     6⋅a ⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k23.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQcAAAA1CAMAAAB7sN6HAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n2XAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{a d_{33}}{6 b} - \\frac{b d_{11}}{6 a} & \\frac{d_{12}}{4} + \\frac{d_{33}}{4}\\\\\\frac{d_{12}}{4} + \\frac{d_{33}}{4} & - \\frac{a d_{11}}{6 b} - \\frac{b d_{33}}{6 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  a⋅d₃₃   b⋅d₁₁     d₁₂   d₃₃   ⎤\n",
       "⎢- ───── - ─────     ─── + ───   ⎥\n",
       "⎢   6⋅b     6⋅a       4     4    ⎥\n",
       "⎢                                ⎥\n",
       "⎢   d₁₂   d₃₃       a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢   ─── + ───     - ───── - ─────⎥\n",
       "⎣    4     4         6⋅b     6⋅a ⎦"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k24.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOcAAAA1CAMAAACN3Aw2AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{a d_{33}}{6 b} - \\frac{b d_{11}}{3 a} & \\frac{d_{12}}{4} - \\frac{d_{33}}{4}\\\\- \\frac{d_{12}}{4} + \\frac{d_{33}}{4} & \\frac{a d_{11}}{6 b} - \\frac{b d_{33}}{3 a}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅d₃₃   b⋅d₁₁    d₁₂   d₃₃  ⎤\n",
       "⎢───── - ─────    ─── - ───  ⎥\n",
       "⎢ 6⋅b     3⋅a      4     4   ⎥\n",
       "⎢                            ⎥\n",
       "⎢   d₁₂   d₃₃   a⋅d₁₁   b⋅d₃₃⎥\n",
       "⎢ - ─── + ───   ───── - ─────⎥\n",
       "⎣    4     4     6⋅b     3⋅a ⎦"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k34.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Vector de fuerzas debido a fuerzas volumétricas. \n",
    "\n",
    "En la expresión del PTV interviene el vector de fuerzas interiores, volumétricas, $\\mathbf{b}$, generando fuerzas nodales equivalentes. Para un elemento: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_b = \\int_{\\Omega} \\mathbf{N}^T \\cdot \\mathbf{b} \\,d \\Omega\n",
    "\\end{equation}\n",
    "\n",
    "La integración genera fuerzas nodales en los cuatro nodos: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_b = \\dfrac{A}{4} \\,\\begin{pmatrix} b_x \\\\b_y \\\\b_x \\\\b_y \\\\ b_x \\\\ b_y \\\\b_x \\\\b_y \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Siendo $A= 4\\,a\\,b$. \n",
    "\n",
    "Si la fuerza por unidad de volumen corresponde al peso propio, entonces $b_x = 0$ y $b_y = - \\rho \\,g$, siendo $\\rho$ la densidad volumétrica del material, y $g$ la constante gravitatoria. \n",
    "\n",
    "Por tanto, para el caso de fuerzas debidas a peso propio, se tomará: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_b = \\dfrac{A}{4} \\,\\begin{pmatrix} 0 \\\\ -\\rho \\,g \\\\ 0 \\\\- \\rho \\,g  \\\\ 0 \\\\ -\\rho \\,g \\\\0\\\\-\\rho\\,g\\end{pmatrix}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Vector de fuerzas debido a tensiones en el contorno\n",
    "\n",
    "Al igual que ocurre con el elemento de 3 nodos, en el caso del elemento de 4 nodos las fuerzas sobre el contorno producen términos de fuerzas nodales en los lados cargados. Por tanto, sólo aparecerán fuerzas nodales en elementos que compartan alguno de sus lados con los lados cargados. \n",
    "\n",
    "Puesto que la geometría es sencilla, consideremos el cálculo para carga constante en el lado $1-2$. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "matN = Matrix.vstack(Nu,Nv)\n",
    "tx,ty = symbols('tx ty')\n",
    "vect = Matrix([[tx,ty]]).T\n",
    "producto = matN.T.multiply(vect)\n",
    "producto = producto.subs(y,-b)\n",
    "f_t = integrate(producto,(x,-a,a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}a tx\\\\a ty\\\\a tx\\\\a ty\\\\0\\\\0\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅tx⎤\n",
       "⎢    ⎥\n",
       "⎢a⋅ty⎥\n",
       "⎢    ⎥\n",
       "⎢a⋅tx⎥\n",
       "⎢    ⎥\n",
       "⎢a⋅ty⎥\n",
       "⎢    ⎥\n",
       "⎢ 0  ⎥\n",
       "⎢    ⎥\n",
       "⎢ 0  ⎥\n",
       "⎢    ⎥\n",
       "⎢ 0  ⎥\n",
       "⎢    ⎥\n",
       "⎣ 0  ⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_t.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Nótese que, teniendo en cuenta que el lado mide \"2a\", se ha obtenido que la fuerza nodal equivalente es la mitad de la resultante. Para carga constante, este resultado es coherente con un reparto isostático en dos puntos. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Del mismo modo, para el lado cargado 1-2, con carga constante: \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACwAAADICAMAAACkj/OnAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAkv6DlDzOPybRSnxGysQSAh23c0ZAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\b tx\\\\b ty\\\\b tx\\\\b ty\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 0  ⎤\n",
       "⎢    ⎥\n",
       "⎢ 0  ⎥\n",
       "⎢    ⎥\n",
       "⎢b⋅tx⎥\n",
       "⎢    ⎥\n",
       "⎢b⋅ty⎥\n",
       "⎢    ⎥\n",
       "⎢b⋅tx⎥\n",
       "⎢    ⎥\n",
       "⎢b⋅ty⎥\n",
       "⎢    ⎥\n",
       "⎢ 0  ⎥\n",
       "⎢    ⎥\n",
       "⎣ 0  ⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "producto = matN.T.multiply(vect)\n",
    "producto = producto.subs(x,a)\n",
    "f_t = integrate(producto,(y,-b,b))\n",
    "f_t.simplify()\n",
    "f_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Para un caso general de tensiones variables, puede demostrarse que este elemento produce unas fuerzas nodales equivalentes a un reparto isostático (es decir, convertir el sistema de vectores deslizantes en dos vectores en los nodos estáticamente equivalentes a la distribución de tensiones original). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Consideración sobre el comportamiento del elemento de 4 nodos\n",
    "\n",
    "Cuando se utiliza un elemento finito, es necesario analizar para qué problemas es adecuado y para cuáles presenta problemas. No debe perderse de vista nunca que el MEF es un método aproximado, y que existen un conjunto de elementos que deben adaptarse a cada aplicación particular. El analista debe seleccionar de forma adecuada aquellos elementos que mejor representan el comportamiento que se desea analizar. \n",
    "\n",
    "El elemento de 4 nodos ofrece una buena solución para los casos en los que el comportamiento sea de tracción-compresión. \n",
    "\n",
    "En problemas en los que el comportamiento predominante sea de flexión, este elemento requiere mallas muy tupidas para obtener resultados aceptables. \n",
    "\n",
    "La mala disponibilidad de este elemento para modelar estados de flexión tiene una explicación muy instructiva. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Para un elemento, la solución exacta con teoría de vigas puede escribirse como: \n",
    "\n",
    "\\begin{align}\n",
    "u(x,y) & = \\dfrac{M}{E\\,I}\\,x\\,y \\\\\n",
    "v(x,y) & = \\dfrac{M\\,a^2}{2\\,E\\,I} \\,\\left( 1 - \\dfrac{x^2}{a^2} \\right) + \\dfrac{M\\,b^2}{2\\,E\\,I}\\,\\left( 1 - \\dfrac{y^2}{b^2} \\right)\n",
    "\\end{align}\n",
    "\n",
    "El elemento rectangular, por tener lados rectos, sólo puede representar el modo de desplazamiento de flexión: \n",
    "\n",
    "\\begin{align}\n",
    "u & = \\overline{u}\\, x\\,y \\\\\n",
    "v & = 0\n",
    "\\end{align}\n",
    "\n",
    "Estas soluciones pueden verse en la siguiente figura (ref. \"Cálculo de estructuras por el método de los elementos finitos, E. Oñate, figura 5.10)\n",
    "\n",
    "<img alt=\"CompFlex\" src=\"comportamiento_elto_rectangular.png\" />\n",
    "\n",
    "El elemento no puede reproducir correctamente la distribución de desplazamientos verticales en un estado de flexión, lo que le confiere una rigidez excesiva. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Adicionalmente, la distribución exacta de deformaciones tangenciales es: \n",
    "\n",
    "\\begin{equation}\n",
    "\\gamma_{xy} = \\dfrac{\\partial u}{\\partial y} + \\dfrac{\\partial v}{\\partial x} = 0\n",
    "\\end{equation}\n",
    "\n",
    "Sin embargo, en la solución aproximada, \n",
    "\n",
    "$$\\gamma_{xy} = \\overline{u}\\,x$$\n",
    "\n",
    "El elemento contiene un \"exceso\" de deformaciones tangenciales. Esto genera una rigidez adicional que le incapacita para reproducir estados de flexión."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Una forma de eliminar este exceso consiste en evaluar $\\gamma_{xy}$ en el centro (donde x=0 y por tanto, $\\gamma_{xy} =0$). Esta es la base del método de _integración reducida_. \n",
    "\n",
    "Existen otras formas de resolver este problema: \n",
    "\n",
    "- Adición de nodos interiores. \n",
    "- Integración reducida de los términos de deformación tangencial. \n",
    "- Adición de modos incompatibles. \n",
    "- Imposición de un campo de deformaciones. \n",
    "\n",
    "En SAP 2000, por ejemplo, pueden encontrarse opciones para Elementos Incompatibles (opción \"Incompatible Modes\"), para mejorar el comportamiento a flexión del elemento. \n",
    "\n",
    "La práctica 4 del curso muestra de forma práctica cómo SAP2000, por defecto, activa una técnica de modos incompatibles para, precisamente, evitar que en su uso para flexión, con pocos elementos, la solución sea excesivamente rígida e incorrecta. El lector de estas líneas podrá comprobar en la práctica qué efecto tiene la activación o desactivación de este recurso."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Para ampliar conocimientos en torno a las diferentes técnicas de mejora del comportamiento del elemento, se recomienda al lector la consulta del apartado 5.4.2 y siguientes del libro del prof. Eugenio Oñate \"Cálculo de Estructuras por el Método de los elementos Finitos\"."
   ]
  }
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