04_MEF_Viga_Timoshenko.ipynb

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    "# El MEF para vigas de Timoshenko\n",
    "\n",
    "\n",
    "Análisis de Estructuras. Grado en Ingeniería Civil. \n",
    "\n",
    "Autores: \n",
    "- Alejandro E. Martínez-Castro, Esther Puertas García, Rafael Gallego.\n",
    "- Dpto. 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: Viga de Timoshenko</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>."
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    "<a id=\"teoria\"></a> \n",
    "## Teoría básica\n",
    "\n",
    "La [viga de Timoshenko](https://es.wikipedia.org/wiki/Teor%C3%ADa_de_vigas_de_Timoshenko) (o viga gruesa) es un modelo de viga que tiene en cuenta la deformación debida a esfuerzo cortante. Es apta para modelizar elementos con canto elevado, para los cuales la relación canto-luz no es adecuada para ser considerados elementos de viga esbeltos (o viga de Bernoulli-Euler). \n",
    "\n",
    "Este modelo de viga comparte las mismas hipótesis que la viga de Bernoulli-Euler, pero difiere en la consideración del giro de la sección. \n",
    "\n",
    "Todos los puntos que están en una sección de $x$ constante (sección plana) permanecen en una sección plana al deformarse la viga. La viga de Bernoulli-Euler plantea la hipótesis de que el giro de esta sección coincide con el de la directriz de la viga en la configuración deformada (este giro coincide con la pendiente para giros pequeños). Esta hipótesis tiene como importante consecuencia que la distorsión angular es cero. En efecto, para calcular la distribución de tensiones tangenciales verticales, la viga de Bernoulli-Euler debe acudir a equilibrar las tensiones (Teorema de Colignon). \n",
    "\n",
    "La viga de Timoshenko admite que la sección plana permanece plana. La principal diferencia con el modelo de viga de Bernoulli-Euler consiste en admitir que el giro de la sección no coincide con la pendiente a la curva directriz. El ángulo girado por la sección es un parámetro independiente. Y constante para toda la sección. En consecuencia, en este modelo, la distorsión angular no es cero. \n",
    "\n",
    "Consideremos un sistema de referencia tal que el eje de la viga coincide con el eje $x$ positivo, el eje $y$ está en el plano horizontal de la sección, y el eje $z$ es perpendicular al plano de la viga. Las cargas se orientan según el eje $z$.  \n",
    "\n",
    "El campo de desplazamientos se notará como $(u,v,w)$, siendo $w(x)$ el desplazamiento vertical. \n",
    "\n",
    "<img alt=\"Secion_Timoshenko\" src=\"hipotesis_timoshenko.png\" />\n",
    "\n",
    "El giro que experimenta una sección plana al deformarse se notará como $\\theta(x)$. Se compone de dos partes: \n",
    "\n",
    "\\begin{equation}\n",
    "\\theta = \\dfrac{d w}{d x} + \\phi\n",
    "\\end{equation}\n",
    "\n",
    "Siendo: \n",
    "+ $\\dfrac{d w}{d x}$ = Giro de la viga de Bernoulli-Euler (la sección se alinea con la normal a la deformada). \n",
    "+ $\\phi$ = $\\phi(x)$, Giro adicional respecto a la normal. \n",
    "+ $\\theta = \\theta (x)$, Giro total de la sección. "
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    "### Desplazamientos y deformaciones.\n",
    "\n",
    "El campo de desplazamientos horizontales es: \n",
    "\\begin{equation}\n",
    "u = -z \\,\\theta\n",
    "\\end{equation}\n",
    "siendo $w$ común para todos los puntos de una misma $x$. \n",
    "\n",
    "Las deformaciones $\\epsilon_x$ y $\\gamma_{xz}$ pueden escribirse como sigue:  \n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\epsilon_x & = &- z \\, \\dfrac{d \\theta}{d x} \\\\\n",
    "\\gamma_{xz} & = &\\dfrac{\\partial w}{\\partial x} + \\dfrac{\\partial u}{\\partial z} = - \\theta + \\dfrac{d w}{d x} = - \\phi\n",
    "\\end{eqnarray}\n",
    "\n",
    "Nótese que en las anteriores expresiones se ha tenido en cuenta que $w$ y $\\theta$ son sólo función de una variable, la $x$. Por eso las derivadas parciales se han escrito como derivadas ordinarias. "
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    "### Tensiones\n",
    "\n",
    "Puesto que $\\sigma_x = E \\, \\epsilon_x$, teniendo en cuenta las expresiones anteriores, \n",
    "\\begin{equation}\n",
    "\\sigma_x = -z \\,E \\,\\dfrac{d \\theta}{d x}\n",
    "\\end{equation}\n",
    "siendo $E$ el módulo de elasticidad. \n",
    "\n",
    "La tensión tangencial $\\tau_{xz} = G \\, \\gamma_{xz}$, con $G = \\dfrac{E}{2\\,(1 + \\nu)}$ el módulo de rigidez a cortante, siendo $\\nu$ el coeficiente de Poisson del material. Teniendo en cuenta la expresión de $\\gamma_{xz}$, \n",
    "\n",
    "\\begin{equation}\n",
    "\\tau_{xz} = G \\, \\left( \\dfrac{d w}{d x} - \\theta \\right) = - G \\,\\phi\n",
    "\\end{equation}\n",
    "\n",
    "Puesto que el ángulo $\\phi$ es el mismo para todos los puntos de una sección de $x$ fija, claramente, el elemento presenta un perfil de tensiones tangenciales constantes, de valor $\\tau_{xz}$. "
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    "### Esfuerzos\n",
    "\n",
    "El esfuerzo cortante $\\mathcal{V}$ se relaciona con $\\tau_{xz}$ de forma \"especial\", ya que _estrictamente_ no se corresponde con la integral de $\\tau_{xz}$ en el área $A$, sino en un área $A_c$ denominada \"área reducida\" o \"área a cortante\" ($A_c = \\alpha \\, A$, siendo $\\alpha$ un coeficiente que depende de la forma de la sección transversal). Así, \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathcal{V} = A_c \\,\\tau_{xz} = A_c \\,G\\,\\gamma_{xz}\n",
    "\\end{equation}\n",
    "\n",
    "Los esfuerzos se relacionan con las funciones $\\theta$ y $w$ como sigue: \n",
    "\n",
    "\\begin{align}\n",
    "\\mathcal{M} & = E\\,I\\,\\dfrac{d \\theta}{d x} \\\\\n",
    "\\mathcal{V} & = G \\,A_c \\cdot \\left( \\dfrac{d w}{d x} - \\theta \\right)\n",
    "\\end{align}\n"
   ]
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    "### Ecuaciones de equilibrio\n",
    "\n",
    "Las ecuaciones de equilibrio de fuerzas verticales y momentos, considerando $q = q(x)$ una fuerza por unidad de línea distribuida en la viga, son: \n",
    "\n",
    "\\begin{align}\n",
    "\\dfrac {d \\mathcal{V}}{d x} + q & = 0 \\\\\n",
    "\\dfrac {d \\mathcal{M}}{d x} + \\mathcal{V} & = 0 \\\\\n",
    "\\end{align}\n",
    "\n",
    "Estas ecuaciones pueden escribirse como: \n",
    "\n",
    "\\begin{align}\n",
    "\\dfrac{d \\mathcal{V}}{d x} & = - q \\Rightarrow G\\,A_c \\cdot \\dfrac{d}{d x} \\,\\left( \\dfrac{d w}{d x} - \\theta \\right) = -q &\\,\\,\\,\\, (1)   \\\\\n",
    "\\dfrac{d \\mathcal{M}}{d x} & = - \\mathcal{V} \\Rightarrow \\dfrac{d}{d x} \\,\\left( E\\,I\\,\\dfrac{d \\theta}{d x}  \\right) = -\\mathcal{V} &\\,\\,\\,\\, (2)\n",
    "\\end{align}\n"
   ]
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    "## Formulación débil.\n",
    "\n",
    "En las ecuaciones de equilibrio, considerando las funciones de ponderación independientes $w^*$ y $\\theta^*$ , estas ecuaciones pueden escribirse como: \n",
    "\n",
    "\\begin{align}\n",
    "G\\,A_c \\cdot \\dfrac{d}{d x} \\,\\left( \\dfrac{d w}{d x} - \\theta \\right)\\cdot w^* = -q \\,\\cdot w^* &\\,\\,\\,\\, \\mathrm{(1a)} \\\\\n",
    "\\dfrac{d}{d x} \\,\\left( E\\,I\\,\\dfrac{d \\theta}{d x}  \\right) \\cdot \\theta^* = -\\mathcal{V}\\,\\theta^* &\\,\\,\\,\\, \\mathrm{(2a)}\n",
    "\\end{align}\n",
    "\n",
    "E integrando entre $x=0$ y $x=L$, \n",
    "\n",
    "\\begin{align}\n",
    "\\int_{0}^{L} G\\,A_c \\cdot \\dfrac{d}{d x} \\,\\left( \\dfrac{d w}{d x} - \\theta \\right)\\cdot w^*\\,dx & =  \\int_{0}^{L} -q \\,\\cdot w^*\\,dx &\\,\\,\\,\\, \\mathrm{(1b)} \\\\\n",
    "\\int_{0}^{L} \\dfrac{d}{d x} \\,\\left( E\\,I\\,\\dfrac{d \\theta}{d x}  \\right) \\cdot \\theta^*\\,dx  & = \\int_{0}^{L} -\\mathcal{V}\\,\\theta^* \\,dx &\\,\\,\\,\\, \\mathrm{(2b)}\n",
    "\\end{align}"
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    "La primera integral de cada igualdad puede integrarse por partes,\n",
    "\n",
    "\\begin{align}\n",
    "\\int_{0}^{L} G\\,A_c \\cdot \\dfrac{d}{d x} \\,\\left( \\dfrac{d w}{d x} - \\theta \\right)\\cdot w^*\\,dx & =  \n",
    "\\left. G \\,A_c \\cdot \\left( \\dfrac{d w}{d x} - \\theta \\right) \\cdot w^* \\right]_{0}^{L} - \\int_{0}^{L} G\\,A_c \\cdot \\left( \\dfrac{d w}{d x} - \\theta \\right) \\cdot \\dfrac{d w^*}{dx}\\,dx \\\\\n",
    "\\int_{0}^{L} \\dfrac{d}{d x} \\,\\left( E\\,I\\,\\dfrac{d \\theta}{d x}\\right) \\cdot \\theta^*\\,dx & = \n",
    "\\left. E I \\dfrac{d \\theta}{d x} \\,\\theta^* \\right]_{0}^{L} - \\int_{0}^{L} E I \\dfrac{d \\theta}{d x} \\cdot \\dfrac{d \\theta^*}{d x}\\,dx\n",
    "\\end{align}"
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    "Nótese que en las igualdades anteriores aparecen evaluaciones en los extremos $0$ y $L$. \n",
    "\n",
    "Considerando el signo positivo de las funciones $\\mathcal{V}$ y $\\mathcal{M}$, podemos escribir: \n",
    "\n",
    "\\begin{align}  \n",
    "\\left. G \\,A_c \\cdot \\left( \\dfrac{d w}{d x} - \\theta \\right) \\cdot w^* \\right]_{0}^{L} & = \\mathcal{V}(L)\\,w^*(L) - \\mathcal{V}(0)\\,w^*(0) = \\\\\n",
    "\\left. E I \\dfrac{d \\theta}{d x} \\,\\theta^* \\right]_{0}^{L} & = \\mathcal{M}(L)\\,\\theta^*(L) - \\mathcal{M}(0)\\,\\theta^*(0)\n",
    "\\end{align}\n"
   ]
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    "Las ecuaciones (1b) y (2b) pueden escribirse como sigue: \n",
    "\n",
    "\\begin{align}\n",
    "\\mathcal{V}(L)\\,w^*(L) - \\mathcal{V}(0)\\,w^*(0) -  \\int_{0}^{L} G\\,A_c \\cdot \\left( \\dfrac{d w}{d x} - \\theta \\right) \\, \\dfrac{d w^*}{dx}\\,dx & = - \\int_{0}^{L} q \\,w^* \\,d x &\\,\\,\\,\\, \\mathrm{(1c)}\\\\\n",
    "\\mathcal{M}(L)\\,\\theta^*(L) - \\mathcal{M}(0)\\,\\theta^*(0) -  \\int_{0}^{L} E\\,I \\cdot \\left( \\dfrac{d \\theta}{d x} \\right) \\, \\dfrac{d \\theta^*}{dx}\\,dx & = - \\int_{0}^{L} G\\, A_c \\, \\left( \\dfrac{d w}{d x} - \\theta \\right) \\, \\theta^* \\, d x &\\,\\,\\,\\, \\mathrm{(2c)}\n",
    "\\end{align}\n",
    "\n",
    "En la expresión anterior (2c) se ha sustituido $\\mathcal{V} = G\\, A_c \\, \\left( \\dfrac{d w}{d x} - \\theta \\right)$."
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    "## Formulación discreta en un elemento.\n",
    "\n",
    "Consideremos que las funciones $w(x)$ y $\\theta(x)$ se aproximan, interpolando a partir de un conjunto de valores discretos $w_j$ y $\\theta_j$, como sigue: \n",
    "\n",
    "\\begin{align}\n",
    "w(x) & \\simeq \\sum_{j=1}^{n} f_j (x) \\, w_j \\\\\n",
    "\\theta (x) & \\simeq \\sum_{j=1}^{n} g_j (x) \\, \\theta_j\n",
    "\\end{align}\n",
    "donde se ha empleado la notación $f_j(x)$ para identificar a las funciones que interpolan desplazamientos (\"flecha\") y $g_j(x)$ a las que interpolan los giros. \n",
    "\n",
    "Si en la ecuación (1c) se eligen como funciones de ponderación $w^* = f_i(x)$, y en la ecuación (2c) $\\theta^* = g_i(x)$, con $i = 1, N$, (es decir, las mismas que se han usado para aproximar $w(x)$ y $\\theta(x)$ ), se tiene: \n",
    "\n",
    "\\begin{align}\n",
    "\\mathcal{V}(L)\\,f_i(L) - \\mathcal{V}(0)\\,f_i(0) + \\int_{0}^{L} q(x)\\,f_i(x)\\,d x & =\n",
    " \\int_{0}^{L} G\\,A_c \\cdot \\left( \\sum_{j=1}^{n} f'_j (x)\\,w_j  - \\sum_{j=1}^{n} g_j(x)\\,\\theta_j \\right)  \\, f'_i(x)\\,dx  &\\,\\,\\,\\, \\mathrm{(1d)}\\\\\n",
    "\\mathcal{M}(L)\\,g_i(L) - \\mathcal{M}(0)\\,g_i(0) + \\int_{0}^{L} G\\, A_c \\, \\left( \\sum_{j=1}^{n} f'_j(x) \\,w_j - \\sum_{j=1}^{n} g_j(x)\\,\\theta_j \\right) \\, g_i(x) \\, d x & = \n",
    "  \\int_{0}^{L} E\\,I \\cdot \\left( \\sum_{n=1}^{n} g'_j(x)\\,\\theta_j \\right) \\, g'_i(x) \\,dx &\\,\\,\\,\\, \\mathrm{(2d)}\n",
    "\\end{align}"
   ]
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    "Las ecuaciones (1d) y (1e) pueden escribirse de forma más compacta como sigue: \n",
    "\n",
    "\\begin{align}\n",
    "\\sum_{j=1}^n \\left[\\begin{matrix}\n",
    "\\int_{0}^{L} G\\,A_c\\,f'_i(x)\\,f'_j(x)\\,dx & - \\int_{0}^{L} G\\,A_c\\,f'_i(x) \\,g_j(x)\\,dx  \\\\\n",
    "-\\int_{0}^{L} G\\,A_c \\,g_i(x)\\,f'_j(x)\\,dx & \\int_{0}^{L} G\\,A_c\\,g_i(x)\\,g_j(x) \\,dx\n",
    "\\end{matrix} \\right] \\cdot \\begin{pmatrix} w_j \\\\ \\theta_j \\end{pmatrix} + \\\\\n",
    "\\sum_{j=1}^n \\left[\\begin{matrix}\n",
    "0 & 0 \\\\\n",
    "0 & \\int_{0}^{L} g'_i(x) \\,E\\,I\\,g'_j(x) \\, dx \\end{matrix} \\right] \\cdot \\begin{pmatrix} w_j \\\\ \\theta_j \\end{pmatrix} =\n",
    "\\begin{pmatrix} \\int_{0}^{L} f_i(x) \\,q(x) \\,dx \\\\ 0 \\end{pmatrix} + \\left[\\begin{matrix} P_1 & M_1 & P_2 & M_2 \\end{matrix} \\right] \\cdot \\begin{pmatrix} f_i(0) \\\\ g_i(0) \\\\ f_i(0) \\\\ g_i(L) \\end{pmatrix}\n",
    "\\end{align}\n",
    "\n",
    "Siendo $P_1 = - \\mathcal{V}(0)$, $P_2 = \\mathcal{V}(L)$, $M_1 = - \\mathcal{M}(0)$ y $M_2 = \\mathcal{M}(L)$, que pueden interpretarse como fuerzas y momentos en los extremos del elemento-barra, aunque su significado es diferente al visto en cálculo matricial. \n",
    "\n",
    "Nótese que en las anteriores expresiones se requieren únicamente derivadas primeras. "
   ]
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   "source": [
    "## Elemento de Timoshenko lagrangiano de 2 nodos\n",
    "\n",
    "El elemento más sencillo para la viga de Timoshenko es el lineal. Tanto los desplazamientos como los giros se interpolan con funciones de forma lineales. "
   ]
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     "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()"
   ]
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   "source": [
    "Las funciones $w(x)$ y $\\theta(x)$ se interpolan como: \n",
    "\\begin{align}\n",
    "w(x) = N_1 (x) \\, w_1 + N_2 (x) \\,w_2 \\\\\n",
    "\\theta(x) = N_1 (x) \\, \\theta_1 + N_2 (x) \\,\\theta_2\n",
    "\\end{align}\n",
    "\n",
    "Que en formato matricial, puede escribirse como sigue: \n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{pmatrix} w(x)\\\\ \\theta(x) \\end{pmatrix} = \n",
    "\\left[ \\begin{matrix} N_1(x) & 0 & N_2(x) & 0 \\\\\n",
    "                      0 & N_1(x) &  0 & N_2(x) \\end{matrix} \\right] \\cdot \n",
    "                      \\begin{pmatrix} w_1 \\\\ \\theta_1 \\\\ w_2 \\\\ \\theta_2 \\end{pmatrix} = \\mathbf{N}\\cdot \\mathbf{a}\n",
    "\\end{equation}"
   ]
  },
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   },
   "source": [
    "A continuación se evaluarán las matrices de deformación a flexión $\\mathbf{B_f}$ y a cortante $\\mathbf{B_c}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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   "source": [
    "from sympy import Matrix\n",
    "E, I, Ac, G, L = symbols('E I Ac G L')\n",
    "N1 = 1 - x / L\n",
    "N2 = x / L\n",
    "funciones_w =    Matrix([[N1, 0, N2, 0]])\n",
    "funciones_giro = Matrix([[0, N1, 0, N2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAyBAMAAAA0MfPNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n0k1tdla7+hJoFoj8j3T/7tPjX57w7FvGp4gnAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 - \\frac{x}{L} & 0 & \\frac{x}{L} & 0\\\\0 & 1 - \\frac{x}{L} & 0 & \\frac{x}{L}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    x         x   ⎤\n",
       "⎢1 - ─    0    ─  0⎥\n",
       "⎢    L         L   ⎥\n",
       "⎢                  ⎥\n",
       "⎢           x     x⎥\n",
       "⎢  0    1 - ─  0  ─⎥\n",
       "⎣           L     L⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "funciones = Matrix.vstack(funciones_w,funciones_giro)\n",
    "funciones"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La matriz $\\mathbf{B_f}$ es la matriz de deformación de flexión, para esta formulación, y se obtiene derivando las funciones de forma que definen $\\theta(x)$. \n",
    "\\begin{equation}\n",
    "\\dfrac{d \\theta}{d x} = \\left[ \\begin{matrix} 0 & \\dfrac{d N_1}{dx} & 0 & \\dfrac{d N_2}{d x} \\end{matrix} \\right] \\cdot \n",
    "\\begin{pmatrix} w_1 \\\\ \\theta_1 \\\\ w_2 \\\\ \\theta_2 \\end{pmatrix} = \\mathbf{B_f}\\,\\mathbf{a}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAbBAMAAABYeKrLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iEJmJZjLNVO+r\nRHY8nXFNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABZ0lEQVQ4Ec2VvUrEQBDH/3gLnqfnV21xrCAI\ngj7AFfcGgoUgXHHY2VjaWJgXEEWwT2sVS0sfwTfwHsBCkbOQA93dTJId9yul22Rm9jc/dpKQALHV\nvY3t6r00oaHxQLOR1RCb8iDM9QbhvXLHEH25gyjZ0gMM/6NHHF9k/vvwdy6XJMLMtZSL7XYel7Q9\nl8BZO49L2p4toBh5Rb1HXnZJIsxcc+A15w1l1jmaT1ndIStCe8RMeTjPmpskTGpP5wu4mTR0OAqT\nxqPO09ITJH1ziTup1m6GtR+93pvzOXPVhPZA3b0ib+hIFCSNZwO4Gnm7Hz4mrO6SRBiPertOGF4n\ni9ZQuuiSRBjPQi6e6lYWrKyzFC5JhPGI87eM81W2PK2i8uqSRBgPZ+3s8MXOfDERCU/ha2U1IhKe\nwGfAMhGR8NwD11aTJyQi4fkGnj3NVomIuKf7idW4pyKGkJH/1+lsvB9/YiXRl3u/fnNq1upg1nYA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & - \\frac{1}{L} & 0 & \\frac{1}{L}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   -1      1⎤\n",
       "⎢0  ───  0  ─⎥\n",
       "⎣    L      L⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bf = funciones_giro.diff(x)\n",
    "Bf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La matriz de deformación a cortante, $\\mathbf{B_c}$ se define como sigue: \n",
    "\n",
    "\\begin{equation}\n",
    "\\dfrac{d w }{d x} - \\theta = \\left[ \\begin{matrix} \\dfrac{d N_1}{dx} & -N_1 & \\dfrac{d N_2}{d x} & - N_2 \\end{matrix} \\right] \\cdot \n",
    "\\begin{pmatrix} w_1 \\\\ \\theta_1 \\\\ w_2 \\\\ \\theta_2 \\end{pmatrix} = \\mathbf{B_c}\\,\\mathbf{a} \n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAN4AAAAbBAMAAADhbCyDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{1}{L} & -1 + \\frac{x}{L} & \\frac{1}{L} & - \\frac{x}{L}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1        x  1  -x ⎤\n",
       "⎢───  -1 + ─  ─  ───⎥\n",
       "⎣ L        L  L   L ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bc = funciones_w.diff(x) - funciones_giro\n",
    "Bc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La Matriz $\\mathbf{K_f}$ se evalúa como \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_f} = \\int_{0}^L E\\,I\\,\\mathbf{B_f}^T \\cdot \\mathbf{B_f}\\,dx\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALgAAABlCAMAAADJTdkEAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & \\frac{E I}{L} & 0 & - \\frac{E I}{L}\\\\0 & 0 & 0 & 0\\\\0 & - \\frac{E I}{L} & 0 & \\frac{E I}{L}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0    0    0    0  ⎤\n",
       "⎢                  ⎥\n",
       "⎢    E⋅I      -E⋅I ⎥\n",
       "⎢0   ───   0  ─────⎥\n",
       "⎢     L         L  ⎥\n",
       "⎢                  ⎥\n",
       "⎢0    0    0    0  ⎥\n",
       "⎢                  ⎥\n",
       "⎢   -E⋅I       E⋅I ⎥\n",
       "⎢0  ─────  0   ─── ⎥\n",
       "⎣     L         L  ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kf = integrate(E* I * Bf.T.multiply(Bf),(x,0,L))\n",
    "Kf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La Matriz $\\mathbf{K_c}$ se evalúa como \n",
    "\n",
    "\\begin{equation}\n",
    "K_c = \\int_{0}^L G\\,A_c\\,\\mathbf{B_c}^T \\cdot \\mathbf{B_c}\\,dx\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASYAAABnCAMAAACaaR+VAAAAV1BMVEX///8AAAAAAAAAAAAAAAAA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\n4tBN3uQ6AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{Ac G}{L} & \\frac{Ac G}{2} & - \\frac{Ac G}{L} & \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{3} L & - \\frac{Ac G}{2} & \\frac{Ac G}{6} L\\\\- \\frac{Ac G}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{L} & - \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{6} L & - \\frac{Ac G}{2} & \\frac{Ac G}{3} L\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ Ac⋅G    Ac⋅G   -Ac⋅G    Ac⋅G ⎤\n",
       "⎢ ────    ────   ──────   ──── ⎥\n",
       "⎢  L       2       L       2   ⎥\n",
       "⎢                              ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L  -Ac⋅G   Ac⋅G⋅L⎥\n",
       "⎢ ────   ──────  ──────  ──────⎥\n",
       "⎢  2       3       2       6   ⎥\n",
       "⎢                              ⎥\n",
       "⎢-Ac⋅G   -Ac⋅G    Ac⋅G   -Ac⋅G ⎥\n",
       "⎢──────  ──────   ────   ──────⎥\n",
       "⎢  L       2       L       2   ⎥\n",
       "⎢                              ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L  -Ac⋅G   Ac⋅G⋅L⎥\n",
       "⎢ ────   ──────  ──────  ──────⎥\n",
       "⎣  2       6       2       3   ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kc = integrate(G * Ac * Bc.T.multiply(Bc),(x,0,L))\n",
    "Kc.simplify()\n",
    "Kc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La formulación completa para el elemento puede escribirse como: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K} \\,\\mathbf{a} = \\begin{pmatrix} P_1 \\\\ M_1 \\\\ P_2 \\\\ M_2 \\end{pmatrix} + \n",
    "\\begin{pmatrix} \\int_{0}^{L} q(x)\\,N_1(x)\\,dx \\\\ 0 \\\\ \\int_{0}^{L} q(x)\\,N_2(x)\\,dx \\\\ 0 \\end{pmatrix}\n",
    "\\end{equation}\n",
    "donde $\\mathbf{K} = \\mathbf{K_c} + \\mathbf{K_f}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nótese que este elemento proporciona una solución continua, tanto en desplazamientos como en giros. La solución es de clase $C^0$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAABnCAMAAAApDfXKAAAAUVBMVEX///8AAAAAAAAAAAAAAAAA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\ncOyQCjqUifewtJ/tOI//ADND/qli5Wg5AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{Ac G}{L} & \\frac{Ac G}{2} & - \\frac{Ac G}{L} & \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{3} L + \\frac{E I}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{6} L - \\frac{E I}{L}\\\\- \\frac{Ac G}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{L} & - \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{6} L - \\frac{E I}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{3} L + \\frac{E I}{L}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ Ac⋅G       Ac⋅G      -Ac⋅G       Ac⋅G    ⎤\n",
       "⎢ ────       ────      ──────      ────    ⎥\n",
       "⎢  L          2          L          2      ⎥\n",
       "⎢                                          ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L   E⋅I  -Ac⋅G   Ac⋅G⋅L   E⋅I⎥\n",
       "⎢ ────   ────── + ───  ──────  ────── - ───⎥\n",
       "⎢  2       3       L     2       6       L ⎥\n",
       "⎢                                          ⎥\n",
       "⎢-Ac⋅G      -Ac⋅G       Ac⋅G      -Ac⋅G    ⎥\n",
       "⎢──────     ──────      ────      ──────   ⎥\n",
       "⎢  L          2          L          2      ⎥\n",
       "⎢                                          ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L   E⋅I  -Ac⋅G   Ac⋅G⋅L   E⋅I⎥\n",
       "⎢ ────   ────── - ───  ──────  ────── + ───⎥\n",
       "⎣  2       6       L     2       3       L ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = Kf + Kc\n",
    "K"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bloqueo de la solución\n",
    "\n",
    "El elemento lagrangiano de 2 nodos proporciona un caso interesante para introducir el problema del bloqueo de la solución, y las técnicas que se han desarrollado para resolver este problema dentro del MEF, como la integración reducida. \n",
    "\n",
    "Consideremos una viga en voladizo, modelizada mediante 1 único elemento. La viga tiene en el nodo inicial condición de empotramiento, y en el nodo final, existe una carga vertical de valor $P$. \n",
    "\n",
    "Con la matriz $\\mathbf{K}$ encontrada anteriormente, es posible plantear el sistema. Eliminando las filas y columnas correspondientes al nodo 1 (pues ambos grados de libertad son nulos), se obtiene el sistema: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_{22}} \\cdot \\begin{pmatrix} w_2 \\\\ \\theta_2 \\end{pmatrix} = \\begin{pmatrix} P \\\\ 0 \\end{pmatrix}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAL4AAAA0CAMAAAADgXIRAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nty72X8sUaAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{Ac G}{L} & - \\frac{Ac G}{2}\\\\- \\frac{Ac G}{2} & \\frac{Ac G}{3} L + \\frac{E I}{L}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ Ac⋅G      -Ac⋅G    ⎤\n",
       "⎢ ────      ──────   ⎥\n",
       "⎢  L          2      ⎥\n",
       "⎢                    ⎥\n",
       "⎢-Ac⋅G   Ac⋅G⋅L   E⋅I⎥\n",
       "⎢──────  ────── + ───⎥\n",
       "⎣  2       3       L ⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K22 = K[2:4,2:4]\n",
    "K22"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La inversa de esta matriz es la matriz de flexibilidad de este sistema: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAR8AAABFBAMAAACBG/9bAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{L^{2}}{\\frac{Ac G}{3} L + \\frac{4 E}{L} I} + \\frac{L}{Ac G} & \\frac{L}{\\frac{Ac G}{6} L + \\frac{2 E}{L} I}\\\\\\frac{L}{\\frac{Ac G}{6} L + \\frac{2 E}{L} I} & \\frac{1}{\\frac{Ac G}{12} L + \\frac{E I}{L}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        2                                ⎤\n",
       "⎢       L            L           L        ⎥\n",
       "⎢──────────────── + ────  ────────────────⎥\n",
       "⎢  ⎛Ac⋅G⋅L   E⋅I⎞   Ac⋅G    ⎛Ac⋅G⋅L   E⋅I⎞⎥\n",
       "⎢4⋅⎜────── + ───⎟         2⋅⎜────── + ───⎟⎥\n",
       "⎢  ⎝  12      L ⎠           ⎝  12      L ⎠⎥\n",
       "⎢                                         ⎥\n",
       "⎢          L                     1        ⎥\n",
       "⎢   ────────────────        ────────────  ⎥\n",
       "⎢     ⎛Ac⋅G⋅L   E⋅I⎞        Ac⋅G⋅L   E⋅I  ⎥\n",
       "⎢   2⋅⎜────── + ───⎟        ────── + ───  ⎥\n",
       "⎣     ⎝  12      L ⎠          12      L   ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A22 = K22.inv()\n",
    "A22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAyBAMAAAC9habhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhB2q7vNmUQi\n3Wapwxo/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8ElEQVQoFWOQ//+JAQr0/39kEHZxhXFDXAwZ\nRGAcEO0I4ab/b6+YFQDnMvxkYOAQhXNZgMYxf4Nz2X4zMLABhaB6OT4wMLADhaBc/gkMDOcXwLn+\nCgzMlxAW5RdpLTuA4PYD7QQBqN6/EB6Uy/ILhcv2GYXL+RWZyzX/RwKEDzUKKgkzeXByWRc9cgC5\nDOpmTgPWO0jchwwMmkjcdgYGe1BwQRV/Z2DYbwDnsgLDan8CnMsDjB5gyMMU8wBlkbhoihmARtkj\njGLoYGB4j2QR0BkrgSbB7OU1YAWnIKgzWNXSHZBkQUwQoGosoCVg1OQNACkmRTEc089RAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}P\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡P⎤\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = symbols('P')\n",
    "vecF = Matrix([[P, 0]]).T\n",
    "vecF"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se resuelve el vector de desplazamientos, multiplicando la inversa de $\\mathbf{K_{22}}$ por el vector $\\mathbf{F}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANoAAABLCAMAAAAh+WkPAAAARVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}P \\left(\\frac{L^{2}}{\\frac{Ac G}{3} L + \\frac{4 E}{L} I} + \\frac{L}{Ac G}\\right)\\\\\\frac{L P}{\\frac{Ac G}{6} L + \\frac{2 E}{L} I}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  ⎛        2              ⎞⎤\n",
       "⎢  ⎜       L            L  ⎟⎥\n",
       "⎢P⋅⎜──────────────── + ────⎟⎥\n",
       "⎢  ⎜  ⎛Ac⋅G⋅L   E⋅I⎞   Ac⋅G⎟⎥\n",
       "⎢  ⎜4⋅⎜────── + ───⎟       ⎟⎥\n",
       "⎢  ⎝  ⎝  12      L ⎠       ⎠⎥\n",
       "⎢                           ⎥\n",
       "⎢           L⋅P             ⎥\n",
       "⎢     ────────────────      ⎥\n",
       "⎢       ⎛Ac⋅G⋅L   E⋅I⎞      ⎥\n",
       "⎢     2⋅⎜────── + ───⎟      ⎥\n",
       "⎣       ⎝  12      L ⎠      ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wsol = A22.multiply(vecF)\n",
    "wsol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La solución aproximada, $w_{2}^{aprox}$ es el primer elemento de este vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO8AAAA/BAMAAAD6YIKlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$P \\left(\\frac{L^{2}}{\\frac{Ac G}{3} L + \\frac{4 E}{L} I} + \\frac{L}{Ac G}\\right)$$"
      ],
      "text/plain": [
       "  ⎛        2              ⎞\n",
       "  ⎜       L            L  ⎟\n",
       "P⋅⎜──────────────── + ────⎟\n",
       "  ⎜  ⎛Ac⋅G⋅L   E⋅I⎞   Ac⋅G⎟\n",
       "  ⎜4⋅⎜────── + ───⎟       ⎟\n",
       "  ⎝  ⎝  12      L ⎠       ⎠"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_aprox = wsol[0]\n",
    "w_aprox"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consideremos ahora una sección rectangular, de base $b$ y altura $h$. Se definen el momento de inercia y el área a cortante. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAA1BAMAAACTu4lQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiUSZq1TvELvdZiIy\nds1Wk1T5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAADsklEQVRIDZ1WS2tTQRT+0jxuHjW5RREEpfcf\nJIW2iKANKApKbVeCblpf4NsLiriRFITiaxHRjQhyd4qgDbh00Yg/oEpABdEGW3Dhphak+Eo9M3Nn\n7txHQuJZzHznO+e78547QMhijX0hrgdiB7b3kB1KHcJciOtIxCx/uJvWL3uSjR5kyDjn9yO9vqak\nY2WSzFbdAvj8Rka0Ov++fwRXT3vMcwkP2YTyZVaUOBfV+dT6ugks8zgv0uQyM46xsjApC2Ce4YDl\nN30k5ofHGicFTpdYXbTdgkY+7zDGbwnm9q1p5CGBp5gQW1TxEO84I6Ky5PJYVbpUZ0zu3OPl8qcR\nYOnIhxpevT2gJUmYeNqoIz/emJQEkr8ZzPLSaNmVmtGqJcoq7AdpO7eKyleDZ4sQbzczw5zkGgpm\n8g/S3GNM2G6Cxvnd4+fYGBcsRsSbqJhUpN4wL9rGartgaPIKyx3lubTixUmajPl6tBQHgUFrCX3a\n0uWblHuC5xccjNkVC5vbqHEBlPALcW3pNtA89P8Scss4w07a0SwbUISVgEdYxRXHi+VIyj5BlnBo\n91AHV2Ne2IdeIHsW+3FdJ6nj6WlOGDt3ALeAgXZTlxvabeNSw9HlN4HUjE50hdW5GwQqNKhebckV\nzFlYaNfbDt+Ui0faqckOedEhde6o53P16JwOrDp3hTIWax0So0P5kWe3eaQwg8He5ZVv2MPlqSbt\nxOgmOrBTddwQ8iomRN6631aC6oKKr2AXcJ7HM9NSHkzv5N9F7o+U9955YxWxaSGv/sfUZVeQMh8z\nPU3dYp1/KFgYrSCj/EQTBctkLp0X2reR9qDtiqQdxLZxTaGEKb/8xfY74mvJWuirsW1b/VzF1I8M\n/euMMt1oPCd8bRhf2JLpRkeGOiDM2D1hI2EjNc39ay7tVXQdid2iKDpumabyFklbRUKsqaNoCYqh\n4dCGj6/IMEieX5NyxSowq5AEtGOz3s1LcrKMR8g0UbdeNSw/c4ouanl1gLVOVjT9OdIzWg7cX7pL\n5f4SEL8Jxgj5fTcYrIx1G6O8ARnhd/yY9IQ8XlJ+APxEYI+lq5RRtGQab70hvVD9iOSOzvJfZEY1\nx+TJEl7qKRreG2ydN+dNPXNfAx81iQ4XaOy6787accmRPHd3YJiNKMoSpnxMiah4nLAnA7fhi8tO\nnu6kdnJsHK6LTFG6j0C30iNdYf6ahLy4upJoSeI1ScRhjewe0gEUFi93L/Iynyg4rlD3oK+pcuOW\ngl2Dy3y9/gHDVPH9k6bd7gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( \\frac{b h^{3}}{12}, \\quad \\frac{5 b}{6} h\\right )$$"
      ],
      "text/plain": [
       "⎛   3       ⎞\n",
       "⎜b⋅h   5⋅b⋅h⎟\n",
       "⎜────, ─────⎟\n",
       "⎝ 12     6  ⎠"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b, h, ld, fi = symbols('b h ld fi')\n",
    "In = Rational(1,12)* b * h**3\n",
    "Acort = Rational(5,6) * b * h\n",
    "In, Acort"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La solución exacta para una viga de Bernoulli-Euler es: \n",
    "\n",
    "\\begin{equation}\n",
    "w_{2}^{BE} = \\dfrac{P \\,L^3}{3\\,E\\,I}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACgAAAAvBAMAAABj1ZXmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{L^{3} P}{3 E I}$$"
      ],
      "text/plain": [
       "  3  \n",
       " L ⋅P\n",
       "─────\n",
       "3⋅E⋅I"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_be = (P* L **3) / (3*E*I)\n",
    "w_be"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se define el ratio $\\varphi$ mediante el cociente entre el desplazamiento aproximado y el exacto para una viga de Bernoulli-Euler. \n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi = \\dfrac{ w_2^{aprox} }{w_2^{BE}}\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQcAAAA/BAMAAAAf2xDgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nbJCTVeq1wyEcnAmBWQ0Ev+7T+BevIT2cJANrIAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{3 E}{L^{3}} I \\left(\\frac{L^{2}}{\\frac{Ac G}{3} L + \\frac{4 E}{L} I} + \\frac{L}{Ac G}\\right)$$"
      ],
      "text/plain": [
       "      ⎛        2              ⎞\n",
       "      ⎜       L            L  ⎟\n",
       "3⋅E⋅I⋅⎜──────────────── + ────⎟\n",
       "      ⎜  ⎛Ac⋅G⋅L   E⋅I⎞   Ac⋅G⎟\n",
       "      ⎜4⋅⎜────── + ───⎟       ⎟\n",
       "      ⎝  ⎝  12      L ⎠       ⎠\n",
       "───────────────────────────────\n",
       "                3              \n",
       "               L               "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi = w_aprox/w_be\n",
    "phi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consideremos ahora los valores de $I$ y $A_c$ para sección rectangular. Al sustituir estos parámetros en la expresión de $\\varphi$, observe que el cociente no depende de $b$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM0AAAA1BAMAAAAQU3IuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nECHynfsv0tZ0wLZ3/+9265GdjP8HSuF/4Er7/GAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{3 E h^{2} \\left(3 E h^{2} + 10 G L^{2}\\right)}{5 G L^{2} \\left(6 E h^{2} + 5 G L^{2}\\right)}$$"
      ],
      "text/plain": [
       "     2 ⎛     2         2⎞\n",
       "3⋅E⋅h ⋅⎝3⋅E⋅h  + 10⋅G⋅L ⎠\n",
       "─────────────────────────\n",
       "      2 ⎛     2        2⎞\n",
       " 5⋅G⋅L ⋅⎝6⋅E⋅h  + 5⋅G⋅L ⎠"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi = phi.subs([(I,In),(Ac,Acort)])\n",
    "phi = phi.simplify()\n",
    "phi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para introducir el parámetro de esbeltez $\\lambda =  \\dfrac{L}{h}$, consideremos que $L = \\lambda \\cdot h$. Emplearemos el símbolo $t$ para denominar a esta esbeltez, para realizar las operaciones simbólicas. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAA1BAMAAACtsiRbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{3 E \\left(3 E + 10 G t^{2}\\right)}{5 G t^{2} \\left(6 E + 5 G t^{2}\\right)}$$"
      ],
      "text/plain": [
       "     ⎛            2⎞ \n",
       " 3⋅E⋅⎝3⋅E + 10⋅G⋅t ⎠ \n",
       "─────────────────────\n",
       "     2 ⎛           2⎞\n",
       "5⋅G⋅t ⋅⎝6⋅E + 5⋅G⋅t ⎠"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = symbols('t')\n",
    "phi = phi.subs(L,t*h)\n",
    "phi = phi.simplify()\n",
    "phi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En la expresión anterior, nótese que $G = \\dfrac{E}{2\\,(1 + \\nu)}$, siendo $\\nu$ el coeficiente de Poisson. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAN8AAAA1BAMAAAAzCCP7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{12 \\left(n + 1\\right) \\left(3 n + 5 t^{2} + 3\\right)}{5 t^{2} \\left(12 n + 5 t^{2} + 12\\right)}$$"
      ],
      "text/plain": [
       "           ⎛         2    ⎞\n",
       "12⋅(n + 1)⋅⎝3⋅n + 5⋅t  + 3⎠\n",
       "───────────────────────────\n",
       "     2 ⎛          2     ⎞  \n",
       "  5⋅t ⋅⎝12⋅n + 5⋅t  + 12⎠  "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = symbols('n')\n",
    "phi = phi.subs(G, E / (2 * (1 + n)))\n",
    "phi = phi.simplify()\n",
    "phi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En la anterior expresión, nótese que para $t\\rightarrow \\infty$, el cociente tiende a 0. Teniendo en cuenta que $t=\\lambda$, se ha obtenido: \n",
    "\n",
    "\\begin{equation}\n",
    "\\lim_{\\lambda \\rightarrow \\infty}\\frac{12 \\left(\\nu + 1\\right) \\left(3 \\nu + 5 \\lambda^{2} + 3\\right)}{5 \\lambda^{2} \\left(12 \\nu + 5 \\lambda^{2} + 12\\right)} = 0\n",
    "\\end{equation}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "El hecho de que este cociente tienda a cero indica que el desplazamiento obtenido mediante la aproximación proporcionada por el elemento de dos nodos tiende a cero a medida que la esbeltez aumenta. En efecto, recordando cómo se ha definido $\\varphi$, el numerador debe hacerse cero a medida que $\\lambda$ aumenta.  \n",
    "\n",
    "El límite puede calcularse de forma simbólica, con el comando de SimPy _limit_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(phi,t,oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El anterior resultado es válido para el rango de coeficientes de Poisson habituales en materiales elásticos. \n",
    "Para un coeficiente de Poisson $\\nu = \\dfrac{1}{4}$ se obtiene:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAA0BAMAAACOQoCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiuzKJ\nRN0MreaOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADH0lEQVRIDe1VTWgTURD+0k3z02xiCupBhUo9\nqAV1kVoQtA1aavWgARuEnnLyakRxpQfbi1BviyBUkVpQQbw0F/EimEsslh6CeFOwCAqVSqOCrVW7\nzvvbvN2mIYJHB7o788188+Zt3vsK1LPyQ6se3AAznXChQbpeKuUkftTDG2CpSvx3g3T9VOhbfXwd\nGs89LgowlQ8kD82dDCBAaBfQDZMG6nWAciBv7MFNgn3WPUSjHAc6gQ4LsaxKbhVOeCfCHqZyEaLs\nB84UcQ54izcSl5RNBZhVVarejDJqMcoiEhfb7/spfVOI/FSl6s0oZEtHT60OtLiuailXaaFV1n13\nQUn+QkhV8xaSQlNFVzigPQQllUFrQUMhKbiCHV90nPmCkgNSJS81Y9vvbfsyi2Pnn9XfSyQL9FU8\nCjlqFSAsNqsl+SpbEHfewdDgGiWa0WDuMkoii4gzg+1aTlJie/Hc0WCPMjlXPo0hI6/lJCVyybiq\nodxtPbY8iFHXXcOdcT2nBivnfFvUS4K+ogTxBvHhBrn/qWa/gPvX1mznf1VnDp/VWzUj/i9xQqM0\nJf4L6kZMMOYG4n9Ddn3C359xj5/v+BQL14t/fG42jbC8Z58AY4HKSFnJooQyWQ6K/y2EVmAUWAki\nXTL/kYfX6cnISvxDeQ7jQhprwAseTIyAi5eZ5SFbj2TZE39FeVUxloGDvKZEFCZerD0Qn6fHIgmX\naEAD5AngRoMhWiE35ozg9fSDfLKULFJIUpYkWa6Jf43SWwL4KD2gVTqAp+3bLKK0jVHjqib+HuXu\nB8rHClRTYhQa/oDrUoRUmu+sJv4eBeY+6jZPf0VGoeGltVTEzkRo2va13badEVGnhUQVOAKiGLX/\nHYzik2W1ym1gOs0pm/v7vw4kq7DkKmwwnywrimsxSnKe13XBLNB8wtj2fbKsKPSDLzli+8AqWjM9\nksG/ok+WFWUcoe/0ZUqscNYdjD8aYx4ztrJPlhUlmRumbUbVNLxYPtiB0U1RBDapp5Qvzo2K6JA7\nNV8dSx0hv81XEkjKwx9AxRULgCrcoB+/yKom8GYX+Q94WvZhi0DTQwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{12 t^{2} + 9}{4 t^{2} \\left(t^{2} + 3\\right)}$$"
      ],
      "text/plain": [
       "   ⎛   2    ⎞\n",
       " 3⋅⎝4⋅t  + 3⎠\n",
       "─────────────\n",
       "   2 ⎛ 2    ⎞\n",
       "4⋅t ⋅⎝t  + 3⎠"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi_particular = phi.subs(n,Rational(1,4)).simplify()\n",
    "phi_particular"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Este cociente claramente tiende a 0 cuando $t$ tiende a $\\infty$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(phi_particular,t,oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conclusión\n",
    "\n",
    "La principal conclusión que se ha obtenido es que, en el límite, el elemento de 2 nodos es incapaz de reproducir el comportamiento exacto de una viga de Timoshenko. \n",
    "\n",
    "Nótese que el hecho de que el cociente haya salido 0 no implica que el desplazamiento sea cero. Únicamente indica que para esbelteces elevadas, el elemento de 2 nodos proporciona un desplazamiento inferior al que debería. No que el desplazamiento en sí tienda a cero. \n",
    "\n",
    "Para aclarar este concepto, observemos la expresión del desplazamiento aproximado."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO8AAAA/BAMAAAD6YIKlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$P \\left(\\frac{L^{2}}{\\frac{Ac G}{3} L + \\frac{4 E}{L} I} + \\frac{L}{Ac G}\\right)$$"
      ],
      "text/plain": [
       "  ⎛        2              ⎞\n",
       "  ⎜       L            L  ⎟\n",
       "P⋅⎜──────────────── + ────⎟\n",
       "  ⎜  ⎛Ac⋅G⋅L   E⋅I⎞   Ac⋅G⎟\n",
       "  ⎜4⋅⎜────── + ───⎟       ⎟\n",
       "  ⎝  ⎝  12      L ⎠       ⎠"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_aprox"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Particularizando para la sección rectangular, e introduciendo el parámetor de esbeltez, se obtiene:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPsAAAA/BAMAAADUJaM3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdqu7zZkQ7zKJVEQi\n3WYRMBq/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHNUlEQVRYCcVYfYgVVRQ/72t25n2nbWYIPj9J\nCFsN/avaR+2WibWrtsUathMq/qO8jUSMtnwIFUKwT3OFINn3R1IUsQ8h+iIcsvqnP95CIljCTgpF\nf6TrZ5of27ln5s7MvTPvY3Vbzx/3nPM7594zc+7H3DMAEoWyEvD/qInu4HE/D4anHF0WOGJyVyA8\n9eDWStCYqenJPUBiICj8gSBQxIZW6ARwLlqb1p4P8ExeDwBFSDPCeYZwzuTboULZ3yvS5sckJGWo\nNxjEuWRuWtX6/a4jRT8mIamKlSLOJXPTqnLe79rhhzzIpjdJiV+xMM49HnXFD2YXvfZh3asxWb0l\nI6LeSSrbHdsNTP/kdsnmSiwPyilnxKrpiLYQHpcRUf+X1CFsR3QAxidBKwBXlidj0bzcuZCTEUFP\nUNbT3QjitiEu2OsqCVqwmABOIQK4xvi5ilfzydoog76GrwBO2tznUxOI9DNTquQ4KNcc0RbaZUDU\noxt+fxTU1Z+uTCy4OI9x0Vxfy6za9xIcaV+eddyediRbuCQDol5YA/OhZWJiHOLjFhftdbXCc9BS\nhhGPz3DRo6CY8KVDtOOp8IiFeKZQdKmtFa5AbCdbMw4ddRNBmNbgyMW+r1qdPVPojNZAyIxC+jpb\nMw4V2hyRhKh9noioq52E5FVLK1RctEkplYf0LcX7gql+sSs+YD3CvvygPgpKPc8gW7gf3z4xDrpj\nDEs7ryplw3G0hDSut9IfJB+ATyRjQ1XFue/X8vGi46lJZ+xIt2MKEsJ5yFRKZFmnZIM86mJLYGs5\nlnvF9UlLK31u/TEjJmj7rN7HzrqjNCtpPQ9C8nHTdU9cYPKMieULOnUm9ZmsnTZKWscMtpGdLGh7\nZdpCU6B/WKviFMTp299lEDplTbz+ZMIEixTCvRiiVbBbn7LINFCj8BQvMg4QpuOAnmYKn6BR+DED\ng2XyANVuFvWuhB8sQfwkiz7t4buKGHTukqG1BnJQpvvtaae16yw2kkIbAZNwx4SrCbSOjmd3dHTk\nUMz4BmQO9ka/SSI2PDzX75g3Wnrs7dXLPMxdSX7Iuj2zZ6g193vfN/gjToo3evsufHvPHcMN/9GF\nnoU3EljzhF7HgG2JUTnsnvfYR+f0ooNzZItHbxi+CLGui1neww0f6gdYSXXEb2jMK75r0NGJHMAP\nJmxCx5rUKPxYUejqHrrhNoASrMblSIHjo4IfKt+8A3TziucEi1jzO+Gj96kb0E+ZOX9psmRXiqjT\nqef2H9O53FJSdRNu4OWXAm82uIHzb1Fg903V5AjjUs2vGAxEwss5+sIsE+IvmvAEYdjs5oLF24tc\nL1QiGPr8DF1beroIymKOO/yLe16Aubqj2kKtmj+6fw56bDqBTZ8Bzlp3J5v695WJYTPS8wsuuxOh\n7uiJUA62/MSeXaCf4QwctxHKJjW1av4weXbpyL4Dq1JEUXFOHDKDe9k6BHgni2TT+UwlVoKxiVHL\nwdsOGnhgHH7SRIyyaRXfrIJlxbdIFD5GN8u3waoU0UG9KHqNOC/ZCdsUs6USy70LqbLHyVlMi6FQ\nYd8pKjsom1ZKU1mr+Pb0QTF8trUIVXqHbRCd11omc9w58CzvwZzFgU2Ppmf0lvJMqKocRO6EH4Cq\njkUP/WehbNopHUInxCWK6Og4jE/GqPCUgmsaKSRdtAt5gjEtaD8MKf0MLIMzmg0y5oQ/iIcCvmmM\n7UvKppVSqvlZVnz0GPBdjn8mrpE5ZpdM3Dc1bkuHLvcsvArqkhKsge8PcjNyJ7w2VALY3/shtlY2\nqfhmtT8rvj09uNhnsG3W2vEMJseu7KM8nO0jPw7v6nInvAvZ2aTim2r+uDQqui4EGK6MsT7DFeiE\nhLVO5JouLk2GN4YlB4anbFLxTbV/QPG9A/e7jjseoAvgljVlAIPdYoAklR1B14MJdkIEXyDGrWzi\neqNdIPw/4cNjnONQLeGiugRwHbaaZJhbJuY21n87V/dJgW9P2axbfP8I6QGqJfbgslwKK6xxH5aH\nHza8iNbqO2w94dU8wF+tOnZg2axffCd7F6HjluU95mfIWk3shPSaxdyW5cell2E9wCpXR8kNr67D\nvxVFpQ1Bls3bKL4TVFl5h4+OerVeOGd/cB3UDQ/4EY4YkEcTy2Z48sV3mJ0ZAqWlg2A9C+IlMXzs\nDTWHVpbN2yi+M93ekUm2jiMOK7uSGzuLXGNcDK+M2YvI69K0fK7ocx3WvdC9WYiWvQAohqOyvPx5\n09UdQ7MC3iRlaikJyHrIGALgUTC8Vvp4pweZnJiQJpr1TovLoVr+suaYGH47KL/WtDcypPxTT7+K\neT9lF1TNGYrBdYlj+GMAD0ho8yr7Je+jQcOFTsEsfa33c+uaUIr3Q9QA38Ek+NRT3goypnFBcdo7\n+yE4cj/XJJ7ceCGr9M42JLhpNZwLdF0UiE49+HfwkNJOC3a6czQ5EDyGUgzGpxhNGjTgf2igOsyT\ngjZvAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$P \\left(\\frac{h^{2} t^{2}}{\\frac{E b h^{2}}{3 t} + \\frac{5 G}{18} b h^{2} t} + \\frac{6 t}{5 G b}\\right)$$"
      ],
      "text/plain": [
       "  ⎛          2  2                 ⎞\n",
       "  ⎜         h ⋅t              6⋅t ⎟\n",
       "P⋅⎜─────────────────────── + ─────⎟\n",
       "  ⎜  ⎛     2          2  ⎞   5⋅G⋅b⎟\n",
       "  ⎜  ⎜E⋅b⋅h    5⋅G⋅b⋅h ⋅t⎟        ⎟\n",
       "  ⎜4⋅⎜────── + ──────────⎟        ⎟\n",
       "  ⎝  ⎝ 12⋅t        72    ⎠        ⎠"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_aprox.subs([(Ac,Acort),(I,In),(L,t*h)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAA1BAMAAAAqg81dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqSzqPaQmMPce+hefcfI7RG5VOgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{12 P t \\left(3 E + 10 G t^{2}\\right)}{5 G b \\left(6 E + 5 G t^{2}\\right)}$$"
      ],
      "text/plain": [
       "       ⎛            2⎞\n",
       "12⋅P⋅t⋅⎝3⋅E + 10⋅G⋅t ⎠\n",
       "──────────────────────\n",
       "       ⎛           2⎞ \n",
       " 5⋅G⋅b⋅⎝6⋅E + 5⋅G⋅t ⎠ "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHcAAAAyBAMAAACUgllmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\infty \\operatorname{sign}{\\left (\\frac{P}{G b} \\right )}$$"
      ],
      "text/plain": [
       "      ⎛ P ⎞\n",
       "∞⋅sign⎜───⎟\n",
       "      ⎝G⋅b⎠"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(_,t,oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el límite, y puesto que el signo del cociente anterior es positivo, para coeficiente de Poisson positivo, se obtiene que el desplazamiento tiende a $\\infty$ a medida que $\\lambda$ tiende a infinito. \n",
    "\n",
    "Esto indica un resultado del todo lógico. A medida que la viga es más y más esbelta, el desplazamiento que se obtiene en el extremo del voladizo aumenta, tendiendo a ser infinito para esbeltez infinita.\n",
    "\n",
    "#### Aclaración.\n",
    "\n",
    "No debe asociarse el término _bloqueo_ de la solución con el hecho de que la función de desplazamiento se bloquee y se obtenga un desplazamiento nulo. El desplazamiento no es nulo, salvo que la carga sea nula. \n",
    "\n",
    "El bloqueo hace referencia a que la solución que se obtiene discrepa de la que debería obtenerse.\n",
    "\n",
    "Un elemento de viga de Timoshenko debe comportarse como uno de Bernoulli-Euler, para esbeltez elevada. Sin ese requisito, este elemento no podría utilizarse para uso general para cualquier esbeltez. \n",
    "\n",
    "La solución exacta para la viga de Timoshenko para el desplazamiento en el extremo 2 es: \n",
    "\n",
    "\\begin{equation}\n",
    "w_2^{Timoshenko} = \\left( \\dfrac{L}{G\\,A_c} + \\dfrac{L^3}{3\\,E\\,I} \\right) \\,P\n",
    "\\end{equation}\n",
    "\n",
    "Con el resultado analítico, se puede comprobar que el cociente\n",
    "\n",
    "\\begin{equation}\n",
    "\\dfrac{w_2^{Timoshenko}}{w_2^{BE}}\n",
    "\\end{equation}\n",
    "\n",
    "tiende a 1 para esbeltez $\\lambda$ tendiendo a infinito. \n",
    "\n",
    "A continuación se comprueba, para la sección rectangular. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAA1BAMAAACtsiRbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ncRKJAQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$P \\left(\\frac{L^{3}}{3 E I} + \\frac{L}{Ac G}\\right)$$"
      ],
      "text/plain": [
       "  ⎛   3        ⎞\n",
       "  ⎜  L      L  ⎟\n",
       "P⋅⎜───── + ────⎟\n",
       "  ⎝3⋅E⋅I   Ac⋅G⎠"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_T = (L / (G * Ac) + L**3 / (3*E*I) )*P\n",
    "w_T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALgAAAA1BAMAAAAAMzzyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWyn9KJ4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{3 E}{L^{3}} I \\left(\\frac{L^{3}}{3 E I} + \\frac{L}{Ac G}\\right)$$"
      ],
      "text/plain": [
       "      ⎛   3        ⎞\n",
       "      ⎜  L      L  ⎟\n",
       "3⋅E⋅I⋅⎜───── + ────⎟\n",
       "      ⎝3⋅E⋅I   Ac⋅G⎠\n",
       "────────────────────\n",
       "          3         \n",
       "         L          "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratio = w_T / w_be\n",
    "ratio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAArBAMAAADlOulrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nc83qk3hk8z9AmbmQ3BZUjwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{3 E}{10 G t^{2}} + 1$$"
      ],
      "text/plain": [
       "  3⋅E      \n",
       "─────── + 1\n",
       "      2    \n",
       "10⋅G⋅t     "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratio = ratio.subs([(Ac,Acort),(I,In),(L,t*h)])\n",
    "ratio = ratio.simplify()\n",
    "ratio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(ratio,t,oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Este ratio indica que los resultados proporcionados por una teoría (viga de Timoshenko) y la otra (viga de Bernoulli-Euler) son coherentes. Cuando una viga de Timoshenko se aplica para esbelteces elevadas, la solución es análoga a la que se obtiene con la teoría de vigas de Bernoulli-Euler. Pero esto es para la solución analítica. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pequeñas esbelteces\n",
    "\n",
    "Para pequeñas esbelteces, se puede comprobar que la solución aproximada tiende a la solución analítica de una viga de Timoshenko. A continuación se verifica tomando como referencia la sección rectangular. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAAsBAMAAABI7D2kAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiuzKJ\nRN0MreaOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADQklEQVRIDe1W0UtTURj/zbnpdNMV1EMQSj0W\n7T5ooBAtuzGJcKP0IV8akVG+OMjYS7D7EkRPK+qhl9yDTwW5B8NHB1ERCEp/QI6KIDAUFJda3n7f\nOXfTxVxMfOjBD87OOd/3O7/7nd+9O98Bqphv4EUW8E6sm+ZiArhh301XQVcKuY4DHfD/ZqyVDI0Z\nDgqVgNV8Hf3LwHngGEHzccAdAwK/qq2oGPOS5RQQzQKzbO44dGIVwTs6hSVlKJYUkEMaaFjaEb1T\nQFhoiwawDq8lY9GnRtMsIkWgYIaYidZnVywtYcqxgQa1WPSp0XQuA1xFOTxo5oD61GqKxRvjstYM\n3DjNwXqtHPxiRd1D8KXV54I31Gd1dyzNMXjT6nNpDit9aqaRXJ7PvO3jy+bajxS2ZalmEk9PoRcp\n2950R2zTfMV/0JPUSq5mmv0F+wpUUsDeC6tE/F/6/INX9iCvD4jsAct3TKdroJkqYd3DZq8vh8Zo\n4RzwA2NzpcjW4OngS06G7XGzjyeGN1q4qmNb5/J1C67LFg/bvIq0GRpQ+j1owB3GrJBL1X7IVixT\njVK7lHV/YXctzdqTU/Nv6hc4rHv3haiBegN17c4x3Em/A8UjWaeMNRX4xKaeBn9Mu4sswAgp8qjf\ncI5hKQ0aClgjcxrt+Sn9aza9x/vaXcpFsTQsKxaWKT98JSgCRpvF2bub2bE8e3SxyaYRyAWy0pez\ncNrCo51l6paKKShwBtNxHvjBpsQse2X6qjJ14IjskOboonLhVKSYH+1Zk1CxTM1hjPs/ifrgovNo\n56oSsm3B0f5i+UoXq3ZCQizjYs2mGc3AI7UwxDaUvOdcVZw88D6Z/JxMjgpW1AU8Mf5QuRiypVtN\nAKhLoC7DiBQwpTk3jca0zLSV5zIkTlZtwx1X+sjUolrtOJvjUL1xeT3zcT6RfdHKWFwxPNNy+A0N\nBXxBlVWrBQTHctygaC5XFf0GNE8ZC7+E21qOSQcKPOaoaQ2eMDqzrlVgcoEObs0vQhVtO4tvYaY/\nr6p24ATjSoVu+w5cE5sxzDyIA53jA9ZRSG2O2OEiBfsiS39o1GrgGZvvSK2YFxcT8EXsS9tw1YdF\nluqof0W7qgP+AMYdL9VYBhVUAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{12 E I}{Ac G L^{2} + 12 E I}$$"
      ],
      "text/plain": [
       "     12⋅E⋅I     \n",
       "────────────────\n",
       "      2         \n",
       "Ac⋅G⋅L  + 12⋅E⋅I"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratio2 = w_aprox / w_T\n",
    "ratio2 = ratio2.simplify()\n",
    "ratio2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAsBAMAAABoLKAcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+hoZdGYIcnwIHsB/ABi+1wEGm51zAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{6 E}{6 E + 5 G t^{2}}$$"
      ],
      "text/plain": [
       "    6⋅E     \n",
       "────────────\n",
       "           2\n",
       "6⋅E + 5⋅G⋅t "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratio2 = ratio2.subs([(Ac,Acort),(I,In),(L,t*h)])\n",
    "ratio2 = ratio2.simplify()\n",
    "ratio2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(ratio2,t,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para pequeñas esbelteces, la solución aproximada tiende a la exacta. Por eso el ratio entre soluciones tiende a 1. \n",
    "\n",
    "El problema se concentra por tanto en la incapacidad del elemento desarrollado para tender a los resultados que debería obtener. \n",
    "\n",
    "El elemento se comporta de forma más rígida a lo que le correspondería. Para esbelteces elevadas, con este elemento se obtendrán desplazamientos que, en realidad, deberían ser mayores. Y esta diferencia es tanto más acusada cuanto mayor es la esbeltez. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integración reducida\n",
    "\n",
    "Existen varias formas de evitar el bloqueo de la solución para grandes esbelteces. \n",
    "\n",
    "Puede demostrarse que el problema del bloqueo procede de la evaluación de la matriz $\\mathbf{K_c}$. En efecto, esta matriz resulta excesivamente rígida para grandes esbelteces. \n",
    "\n",
    "Una forma de mejorar el comportamiento de este elemento consiste en sub-integrar la integral que define $\\mathbf{K_c}$. \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_c} = \\int_{0}^{L} G\\,A_c\\,\\mathbf{B_c}^T\\cdot\\mathbf{B_c}\\,dx\n",
    "\\end{equation}\n",
    "\n",
    "Veamos en primer lugar qué orden en $x$ tiene el producto $\\mathbf{B_c}^T\\cdot\\mathbf{B_c}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAisAAABvCAMAAAAaJnBJAAAAVFBMVEX///8AAAAAAAAAAAAAAAAA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\nmJfRefFxb/IAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{L^{2}} & - \\frac{1}{L} \\left(-1 + \\frac{x}{L}\\right) & - \\frac{1}{L^{2}} & \\frac{x}{L^{2}}\\\\- \\frac{1}{L} \\left(-1 + \\frac{x}{L}\\right) & \\left(-1 + \\frac{x}{L}\\right)^{2} & \\frac{1}{L} \\left(-1 + \\frac{x}{L}\\right) & - \\frac{x}{L} \\left(-1 + \\frac{x}{L}\\right)\\\\- \\frac{1}{L^{2}} & \\frac{1}{L} \\left(-1 + \\frac{x}{L}\\right) & \\frac{1}{L^{2}} & - \\frac{x}{L^{2}}\\\\\\frac{x}{L^{2}} & - \\frac{x}{L} \\left(-1 + \\frac{x}{L}\\right) & - \\frac{x}{L^{2}} & \\frac{x^{2}}{L^{2}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡              ⎛     x⎞                        ⎤\n",
       "⎢             -⎜-1 + ─⎟                        ⎥\n",
       "⎢    1         ⎝     L⎠     -1          x      ⎥\n",
       "⎢    ──       ──────────    ───         ──     ⎥\n",
       "⎢     2           L           2          2     ⎥\n",
       "⎢    L                       L          L      ⎥\n",
       "⎢                                              ⎥\n",
       "⎢ ⎛     x⎞                      x     ⎛     x⎞ ⎥\n",
       "⎢-⎜-1 + ─⎟            2    -1 + ─  -x⋅⎜-1 + ─⎟ ⎥\n",
       "⎢ ⎝     L⎠    ⎛     x⎞          L     ⎝     L⎠ ⎥\n",
       "⎢──────────   ⎜-1 + ─⎟     ──────  ────────────⎥\n",
       "⎢    L        ⎝     L⎠       L          L      ⎥\n",
       "⎢                                              ⎥\n",
       "⎢                    x                         ⎥\n",
       "⎢               -1 + ─                         ⎥\n",
       "⎢   -1               L       1         -x      ⎥\n",
       "⎢   ───         ──────       ──        ───     ⎥\n",
       "⎢     2           L           2          2     ⎥\n",
       "⎢    L                       L          L      ⎥\n",
       "⎢                                              ⎥\n",
       "⎢               ⎛     x⎞                       ⎥\n",
       "⎢            -x⋅⎜-1 + ─⎟                 2     ⎥\n",
       "⎢    x          ⎝     L⎠    -x          x      ⎥\n",
       "⎢    ──      ────────────   ───         ──     ⎥\n",
       "⎢     2           L           2          2     ⎥\n",
       "⎣    L                       L          L      ⎦"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bc.T.multiply(Bc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En este producto las potencias más altas de $x$ son cuadráticas. \n",
    "\n",
    "Anteriormente se evaluaron las integrales de forma analítica. Estas integrales pueden evaluarse de forma numérica, mediante cuadraturas de Gauss-Legendre. \n",
    "\n",
    "Una cuadratura de Gauss-Legendre de orden $n$ integra de forma exacta un polinomio de grado $2\\,n - 1$. \n",
    "\n",
    "Con $n=1$, se integra de forma exacta un polinomio de grado $2\\cdot 1 - 1 = 1$. Para integrar de forma exacta se requiere por tanto $n=2$, que proporciona una integral exacta para polinomios de grado $2\\,2 -1 = 3$. En este caso el grado más alto es el cuadrático. Por tanto, $\\mathbf{K_c}$ se puede integrar de forma exacta con una cuadratura de orden 2. \n",
    "\n",
    "Si se toma $n=1$, la integral será \"imperfecta\". Puede demostrarse que tomando $n=1$, estas integrales se sub-evalúan. Es decir, el área que se evalúa (que en esencia, es el sentido de una integral) es menor que la que se evaluaría integrando de forma exacta. \n",
    "\n",
    "Esta forma de evaluar las integrales mediante una aproximación con menor número de puntos de Gauss se denomina _integración reducida_.\n",
    "\n",
    "Para evaluar con 1 punto de Gauss, y en referencia a una función $f(x)$, se puede escribir: \n",
    "\n",
    "\\begin{equation}\n",
    "\\int_0^L f(x)\\,dx = \\int_{-1}^{1} f(\\xi)\\,\\dfrac{d x}{d\\xi} \\,d(\\xi) \\simeq f(\\xi_1)\\cdot W_1 \\cdot \\dfrac{L}{2}\n",
    "\\end{equation}\n",
    "\n",
    "Siendo $\\xi_1$ y $W_1$ la abscisa y el peso de una cuadratura de Gauss de orden 1. Este valor es $\\xi_1 = 1$, y $W_1 = 2$. Además, se ha tenido en cuenta la relación lineal entre la variable $x$ y $\\xi$, en \n",
    "\n",
    "$$\\dfrac{d x}{d \\xi} = \\dfrac{L}{2}$$.\n",
    "\n",
    "Teniendo en cuenta que $\\xi_1 = 0$, lo cual implica que $x_1 = \\dfrac{L}{2}$, puede escribirse: \n",
    "\n",
    "\\begin{equation}\n",
    "\\int_0^L f(x)\\,dx \\simeq f(\\dfrac{L}{2}) \\cdot 2 \\,\\cdot \\dfrac{L}{2} = f(\\dfrac{L}{2}) \\cdot \\,L\n",
    "\\end{equation}\n",
    "\n",
    "La expresión anterior puede verificarse fácilmente tomando $f(x) = 1$. Claramente, \n",
    "\n",
    "\\begin{equation}\n",
    "\\int_0^L 1 \\,dx = L \\simeq f(\\dfrac{L}{2}) \\cdot L = L\n",
    "\\end{equation}\n",
    "\n",
    "En relación al cálculo de $\\mathbf{K_c}$, la aproximación con 1 punto de gauss es: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_c} \\simeq G\\,A_c\\,\\mathbf{\\hat{B}_c}\\cdot \\mathbf{\\hat{B}_c} \\cdot L\n",
    "\\end{equation}\n",
    "\n",
    "siendo $\\mathbf{\\hat{B}_c}$ la matriz $\\mathbf{B_c}$ evaluada en $x=\\dfrac{L}{2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASYAAABnCAMAAACaaR+VAAAAV1BMVEX///8AAAAAAAAAAAAAAAAA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\ni6xGBseg+g8HMON2w8XwZAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{Ac G}{L} & \\frac{Ac G}{2} & - \\frac{Ac G}{L} & \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{4} L & - \\frac{Ac G}{2} & \\frac{Ac G}{4} L\\\\- \\frac{Ac G}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{L} & - \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{4} L & - \\frac{Ac G}{2} & \\frac{Ac G}{4} L\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ Ac⋅G    Ac⋅G   -Ac⋅G    Ac⋅G ⎤\n",
       "⎢ ────    ────   ──────   ──── ⎥\n",
       "⎢  L       2       L       2   ⎥\n",
       "⎢                              ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L  -Ac⋅G   Ac⋅G⋅L⎥\n",
       "⎢ ────   ──────  ──────  ──────⎥\n",
       "⎢  2       4       2       4   ⎥\n",
       "⎢                              ⎥\n",
       "⎢-Ac⋅G   -Ac⋅G    Ac⋅G   -Ac⋅G ⎥\n",
       "⎢──────  ──────   ────   ──────⎥\n",
       "⎢  L       2       L       2   ⎥\n",
       "⎢                              ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L  -Ac⋅G   Ac⋅G⋅L⎥\n",
       "⎢ ────   ──────  ──────  ──────⎥\n",
       "⎣  2       4       2       4   ⎦"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kc_reducida = G * Ac * Bc.T.multiply(Bc) \n",
    "Kc_reducida = Kc_reducida.subs(x,L/2) * L\n",
    "Kc_reducida "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Formemos de nuevo la matriz $\\mathbf{K}$, pero en esta ocasión tomando la aproximación de $\\mathbf{K_c}$ con 1 punto de gauss. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAABnCAMAAAApDfXKAAAAUVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{Ac G}{L} & \\frac{Ac G}{2} & - \\frac{Ac G}{L} & \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{4} L + \\frac{E I}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{4} L - \\frac{E I}{L}\\\\- \\frac{Ac G}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{L} & - \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{4} L - \\frac{E I}{L} & - \\frac{Ac G}{2} & \\frac{Ac G}{4} L + \\frac{E I}{L}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ Ac⋅G       Ac⋅G      -Ac⋅G       Ac⋅G    ⎤\n",
       "⎢ ────       ────      ──────      ────    ⎥\n",
       "⎢  L          2          L          2      ⎥\n",
       "⎢                                          ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L   E⋅I  -Ac⋅G   Ac⋅G⋅L   E⋅I⎥\n",
       "⎢ ────   ────── + ───  ──────  ────── - ───⎥\n",
       "⎢  2       4       L     2       4       L ⎥\n",
       "⎢                                          ⎥\n",
       "⎢-Ac⋅G      -Ac⋅G       Ac⋅G      -Ac⋅G    ⎥\n",
       "⎢──────     ──────      ────      ──────   ⎥\n",
       "⎢  L          2          L          2      ⎥\n",
       "⎢                                          ⎥\n",
       "⎢ Ac⋅G   Ac⋅G⋅L   E⋅I  -Ac⋅G   Ac⋅G⋅L   E⋅I⎥\n",
       "⎢ ────   ────── - ───  ──────  ────── + ───⎥\n",
       "⎣  2       4       L     2       4       L ⎦"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Kred = Kf + Kc_reducida\n",
    "Kred"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAAwBAMAAADtMzlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnRkhxun+A44b9t/E73jYAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{L^{3} P}{4 E I} + \\frac{L P}{Ac G}$$"
      ],
      "text/plain": [
       "  3         \n",
       " L ⋅P   L⋅P \n",
       "───── + ────\n",
       "4⋅E⋅I   Ac⋅G"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Krig_red = Kred[2:4,2:4]\n",
    "wred = Krig_red.inv().multiply(vecF)\n",
    "w2_red = wred[0].simplify()\n",
    "w2_red"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparemos la expresión de la solución que se obtuvo anteriormente, y la que se ha obtenido con integración reducida"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaEAAAA/BAMAAAC7nWp2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTxXuC3YkXyAMihBKKiCopQr3BUvJNwHSLVj0no9vuwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( P \\left(\\frac{L^{2}}{\\frac{Ac G}{3} L + \\frac{4 E}{L} I} + \\frac{L}{Ac G}\\right), \\quad \\frac{L^{3} P}{4 E I} + \\frac{L P}{Ac G}\\right )$$"
      ],
      "text/plain": [
       "⎛  ⎛        2              ⎞    3         ⎞\n",
       "⎜  ⎜       L            L  ⎟   L ⋅P   L⋅P ⎟\n",
       "⎜P⋅⎜──────────────── + ────⎟, ───── + ────⎟\n",
       "⎜  ⎜  ⎛Ac⋅G⋅L   E⋅I⎞   Ac⋅G⎟  4⋅E⋅I   Ac⋅G⎟\n",
       "⎜  ⎜4⋅⎜────── + ───⎟       ⎟              ⎟\n",
       "⎝  ⎝  ⎝  12      L ⎠       ⎠              ⎠"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_aprox,w2_red"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Es interesante comparar la expresión de la solución exacta de la viga de Timoshenko con la que se ha obtenido mediante integración reducida"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAU8AAAA1BAMAAAAzLXTSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nFArivwWJgdWJGs7lAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( P \\left(\\frac{L^{3}}{3 E I} + \\frac{L}{Ac G}\\right), \\quad \\frac{L^{3} P}{4 E I} + \\frac{L P}{Ac G}\\right )$$"
      ],
      "text/plain": [
       "⎛  ⎛   3        ⎞    3         ⎞\n",
       "⎜  ⎜  L      L  ⎟   L ⋅P   L⋅P ⎟\n",
       "⎜P⋅⎜───── + ────⎟, ───── + ────⎟\n",
       "⎝  ⎝3⋅E⋅I   Ac⋅G⎠  4⋅E⋅I   Ac⋅G⎠"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_T,w2_red"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para valorar el efecto que ha tenido esta integración reducida, es interesante ver su efecto en el cociente\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi = \\dfrac{ w_2^{aprox} }{w_2^{BE}}\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAArBAMAAABiCwBtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{3}{4} + \\frac{3 E I}{Ac G L^{2}}$$"
      ],
      "text/plain": [
       "3    3⋅E⋅I \n",
       "─ + ───────\n",
       "4         2\n",
       "    Ac⋅G⋅L "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratio_fi = w2_red / w_be\n",
    "ratio_fi = ratio_fi.simplify()\n",
    "ratio_fi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para sección rectangular, e introduciendo la esbeltez, se obtiene: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAArBAMAAAB1KZCkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYGJ+wxwA/xr6CVK5ymN89ZrpAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{3 E}{10 G t^{2}} + \\frac{3}{4}$$"
      ],
      "text/plain": [
       "  3⋅E     3\n",
       "─────── + ─\n",
       "      2   4\n",
       "10⋅G⋅t     "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratio_fi = ratio_fi.subs([(I,In),(Ac,Acort),(L,t*h)])\n",
    "ratio_fi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Y el límite para esbeltez tendiendo a infinito es,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAArBAMAAABcLm8jAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAmElEQVQYGWNgEFIyYWBgYAxg8E9gYGD/wsDf\nwMDAuZJh/wGgKANYkIGBuwnEZtSIBlEMDFoTwBTXaqCYAAPzFwYG/t9gisuAgeMjAwNbAoN/AVDJ\n1FBLsELyiP8g8IFIvVch6prBFPtiMCV2GEwpgCk2ATC1nQFMKYAplgtgahMDmPJ79+7Xc7COxWCS\n4TOYUv7/DMJlYAAAAZMma63EH5YAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{3}{4}$$"
      ],
      "text/plain": [
       "3/4"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(ratio_fi,t,oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se ha obtenido un valor de $\\dfrac{3}{4} = 0.75$. Nótese que este valor está próximo a 1. \n",
    "\n",
    "La integración reducida ha permitido que el problema de bloqueo de la solución desaparezca. El resultado sigue siendo imperfecto. Este resultado puede mejorarse y hacerse más próximo a 1 tomando más de 1 elemento para la solución. \n",
    "\n",
    "Tomemos 2 elementos, y nombremos los nodos como 1-2-3 (1 = nodo en el empotramiento, 2 = nodo central, 3 = nodo en el voladizo). Denominemos \"barra a\" y \"barra b\" a las dos barras. Formulando el sistema, y eliminando las filas y columnas correspondientes a los desplazamientos $w_1$, $\\theta_1$, pues son cero por estar en un empotramiento, se obtiene:  \n",
    "\n",
    "\\begin{equation}\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "\\mathbf{K_{22}}^{a+b} & \\mathbf{K_{23}}^b \\\\\n",
    "\\mathbf{K_{32}}^b & \\mathbf{K_{33}}^b\n",
    "\\end{matrix} \\right] \\cdot \n",
    "\\begin{pmatrix}\n",
    "w_2 \\\\ \\theta_2 \\\\ w_3 \\\\ \\theta_3 \\end{pmatrix} = \n",
    "\\begin{pmatrix} 0 \\\\ 0 \\\\ P \\\\ 0 \\end{pmatrix}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "K22a = Kred.subs(L,L/2)[2:4,2:4]\n",
    "K22b = Kred.subs(L,L/2)[0:2,0:2]\n",
    "K33b = K22a\n",
    "K23b = Kred.subs(L,L/2)[0:2,2:4]\n",
    "K22 = K22a + K22b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANUAAAA0CAMAAAAqho2WAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nhUmizK28q9naahV7Lnlv8R/RUDJ7Fgn8VAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{2 Ac}{L} G & \\frac{Ac G}{2}\\\\- \\frac{Ac G}{2} & \\frac{Ac G}{8} L - \\frac{2 E}{L} I\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-2⋅Ac⋅G        Ac⋅G     ⎤\n",
       "⎢────────       ────     ⎥\n",
       "⎢   L            2       ⎥\n",
       "⎢                        ⎥\n",
       "⎢ -Ac⋅G    Ac⋅G⋅L   2⋅E⋅I⎥\n",
       "⎢ ──────   ────── - ─────⎥\n",
       "⎣   2        8        L  ⎦"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K23b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAbYAAABnCAMAAABbwpd8AAAARVBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{4 Ac}{L} G & 0 & - \\frac{2 Ac}{L} G & \\frac{Ac G}{2}\\\\0 & \\frac{Ac G}{4} L + \\frac{4 E}{L} I & - \\frac{Ac G}{2} & \\frac{Ac G}{8} L - \\frac{2 E}{L} I\\\\- \\frac{2 Ac}{L} G & - \\frac{Ac G}{2} & \\frac{2 Ac}{L} G & - \\frac{Ac G}{2}\\\\\\frac{Ac G}{2} & \\frac{Ac G}{8} L - \\frac{2 E}{L} I & - \\frac{Ac G}{2} & \\frac{Ac G}{8} L + \\frac{2 E}{L} I\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 4⋅Ac⋅G                   -2⋅Ac⋅G        Ac⋅G     ⎤\n",
       "⎢ ──────         0         ────────       ────     ⎥\n",
       "⎢   L                         L            2       ⎥\n",
       "⎢                                                  ⎥\n",
       "⎢          Ac⋅G⋅L   4⋅E⋅I   -Ac⋅G    Ac⋅G⋅L   2⋅E⋅I⎥\n",
       "⎢   0      ────── + ─────   ──────   ────── - ─────⎥\n",
       "⎢            4        L       2        8        L  ⎥\n",
       "⎢                                                  ⎥\n",
       "⎢-2⋅Ac⋅G       -Ac⋅G        2⋅Ac⋅G       -Ac⋅G     ⎥\n",
       "⎢────────      ──────       ──────       ──────    ⎥\n",
       "⎢   L            2            L            2       ⎥\n",
       "⎢                                                  ⎥\n",
       "⎢  Ac⋅G    Ac⋅G⋅L   2⋅E⋅I   -Ac⋅G    Ac⋅G⋅L   2⋅E⋅I⎥\n",
       "⎢  ────    ────── - ─────   ──────   ────── + ─────⎥\n",
       "⎣   2        8        L       2        8        L  ⎦"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Krig = zeros(4,4)\n",
    "Krig[0:2,0:2] = K22\n",
    "Krig[2:4,2:4] = K33b\n",
    "Krig[0:2,2:4] = K23b\n",
    "Krig[2:4,0:2] = K23b.T\n",
    "Krig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se forma el vector de la derecha, y se resuelve el sistema de ecuaciones, mediante un procedimiento análogo al visto anteriormente (empleando la inversa de la matriz del sistema). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAAwBAMAAADHg8jeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYloc+AfosB7WuWXr7AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{5 L^{3} P}{16 E I} + \\frac{L P}{Ac G}$$"
      ],
      "text/plain": [
       "   3         \n",
       "5⋅L ⋅P   L⋅P \n",
       "────── + ────\n",
       "16⋅E⋅I   Ac⋅G"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecF = Matrix([[0,0,P,0]]).T\n",
    "w3 = Krig.inv().multiply(vecF)\n",
    "w3 = w3[2]\n",
    "w3 = w3.simplify()\n",
    "w3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se evalúa el ratio entre esta solución y la solución de la viga de Bernoulli-Euler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAArBAMAAACJPLtuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGCk3Fr8AFmb+mmQ7TxwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{3 E}{10 G t^{2}} + \\frac{15}{16}$$"
      ],
      "text/plain": [
       "  3⋅E     15\n",
       "─────── + ──\n",
       "      2   16\n",
       "10⋅G⋅t      "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ratio3 = w3 /w_be\n",
    "ratio3 = ratio3.subs([(Ac,Acort),(I,In),(L,t*h)])\n",
    "ratio3 = ratio3.simplify()\n",
    "ratio3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArBAMAAABlSd54AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYyiSK7\nRN2jbGmSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVQoFWNgAIMyg6kCIIaQAQOD/v9fQBaz\nqz+QXeW2ASyfD2Q7gFkMDOhsj7ZIuJrDDPsvgDggNQwMPA9AJITN+hvKZilg4PoLZTMVMLB+hrJZ\nGRj4CqBs7gCGpg1Adoh+SQPD1iXBIGEag/9w8IEaNq3KmgAyBhSe5hdYC2DheZSB5wBIHBgmvOAA\ngbB5FECCQAAU569cFgtj21cwMAlAxe0/M7Aqw9Q8YOCAhSEw0Dh+QMVZFBDinED1ClBxBi8GI5A5\n4PDkCvUECSMAAD0OPj+sBMVhAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{15}{16}$$"
      ],
      "text/plain": [
       "15\n",
       "──\n",
       "16"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(ratio3,t,oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAOBAMAAACIi8KdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQklEQVQYGT2Qu0oDURCGv73kflmJjdhkE7EX\nV0HEYmsLCfoArijYBcEHSMDCwhSLdjaJIIiFuFZWwhZa2EjwAcTKMrjeQiC6zongFOc/O9/OP3MG\nSnPzqCg593DkXJNoTTkO9bHVHdhmwhemnXLsZ2warhnH8ZDL+B3MDoYnMN8hbxf7WE1d/g3ZfBDJ\nhaQigZaH8Vq4oOwZYNQEq2RIoS9aDih+ijZcOZ5UrUTZpvAjmpXKb2m9p5IB3K5MQ3uGhLQWT5If\nZBYq6u7CFe0ubfsPUmdX2S92xU1Upmn+22L21lTP7Am8jKA+UAOlVIVEPsr45OTjQO6jEXIBpnqK\nRDK0IgW14Wg+PVJL0D0h5jnrfjYgPSDxBjqkJb3PZE37orihbWG6NJ5JCZRFLsuKxnt30ILZqsz5\nWL0RcCZOh0sVfgFK/Fi5CMSecAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$0.9375$$"
      ],
      "text/plain": [
       "0.9375"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "15 / 16"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nótese que al emplear 2 elementos, la formulación con matriz de cortante sub-evaluada con 1 punto de Gauss proporciona una mejor aproximación al cociente. Con 2 elementos el resultado es más próximo a 1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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