EQE512-Week07-Lecture.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h1 align=\"center\">EQE512 MATRIX METHODS IN STRUCTURAL ANALYSIS \n",
    "<br>\n",
    "<br>\n",
    "Week 07\n",
    "<br>\n",
    "<br>\n",
    "Frame Systems- Compression/Tension Members - Support displacement</h1> \n",
    "\n",
    "<h3 align=\"center\">Dr. Ahmet Anıl Dindar (adindar@gtu.edu.tr)</h3> \n",
    "<h4 align=\"center\">2021 Fall  </h4> "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Today's Topics**\n",
    "\n",
    "1- Frame member stiffness matrix\n",
    "\n",
    "2- Frame System Stiffness matrix\n",
    "\n",
    "3- Support settlements\n",
    "\n",
    "4- Elastic supports"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Frame member stiffness matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Freedom**\n",
    "\n",
    "<img src=\"figures/9-EQE512-ColumnMemberExplanation.PNG\" width=\"1000\">\n",
    "\n",
    "---\n",
    "\n",
    "**Transformation Matrix**\n",
    "\n",
    "Remember the relation between local and global terms\n",
    "\n",
    "$$ {\\large k^i = (T^i) \\hat{k}^i T^i }$$\n",
    "\n",
    "Remember the transformation matrix;\n",
    "\n",
    "$$ {\\large\n",
    "T^i = \\begin{bmatrix} \n",
    "\\frac{\\Delta_1}{L} & \\frac{\\Delta_2}{L}  & 0 & 0 & 0 & 0 \\\\ \n",
    "-\\frac{\\Delta_2}{L} & \\frac{\\Delta_1}{L}  & 0 & 0 & 0 & 0  \\\\ \n",
    "0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 &\\frac{\\Delta_1}{L} & \\frac{\\Delta_2}{L}  & 0  \\\\\n",
    "0 & 0 & 0 & -\\frac{\\Delta_2}{L} & \\frac{\\Delta_1}{L}  & 0   \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 1 \n",
    "\\end{bmatrix}}$$\n",
    "\n",
    "---\n",
    "\n",
    "**Local Stiffness Matrix**\n",
    "\n",
    "<img src=\"figures/column_stiffness_in_local_coordinates.png\" width=\"600\">\n",
    "\n",
    "**Global Stiffness Matrix**\n",
    "\n",
    "After applying the transformation matrix\n",
    "\n",
    "<img src=\"figures/9-EQE512-ColumnMemberStiffness.PNG\" width=\"900\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Transformation Matrix of Frame Element**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def transformation_matrix( x1,y1,x2,y2) :\n",
    "    delta_1 = (x2-x1)\n",
    "    delta_2 = (y2-y1)\n",
    "    L = ( (delta_1)**2 + (delta_2)**2)**(0.5)\n",
    "    \n",
    "    T = np.array( [[ delta_1 / L , delta_2 / L  , 0 , 0 , 0 , 0],\n",
    "                   [-delta_2 / L , delta_1 / L  , 0 , 0 , 0 , 0],\n",
    "                   [0 , 0 , 1 , 0 , 0 , 0  ], \n",
    "                   [ 0 , 0 , 0 , delta_1 / L , delta_2 / L , 0 ],\n",
    "                   [ 0 , 0 , 0, -delta_2 / L , delta_1 / L  , 0],\n",
    "                   [0 , 0 , 0 , 0 , 0 , 1] ])\n",
    "    \n",
    "    return( T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Frame Element Member Stiffness Matrix in Local Coordinates**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def column_stiffness_matrix( E , A , I , x1,y1,x2,y2):\n",
    "    delta_1 = (x2-x1)\n",
    "    delta_2 = (y2-y1)\n",
    "    L = ( (delta_1)**2 + (delta_2)**2)**(0.5)\n",
    "    #\n",
    "    EA_L = E * A / L\n",
    "    EI_L = E * I / L\n",
    "    EI_L2 = EI_L / L\n",
    "    EI_L3 = EI_L2 / L\n",
    "    #\n",
    "    k = np.array([[EA_L , 0 , 0 , -EA_L , 0 , 0],\n",
    "                  [ 0 , 12*EI_L3 , 6*EI_L2 ,0 , -12*EI_L3 , 6*EI_L2 ],\n",
    "                  [ 0 , 6*EI_L2 , 4*EI_L , 0 , -6*EI_L2 , 2*EI_L ],\n",
    "                  [ -EA_L , 0 , 0 , EA_L , 0 , 0 ],\n",
    "                  [ 0 , -12*EI_L3 ,-6*EI_L2 , 0 , 12*EI_L3 , -6*EI_L2 ],\n",
    "                  [ 0 , 6*EI_L2 , 2*EI_L , 0 , -6*EI_L2 , 4*EI_L]])\n",
    "                  \n",
    "    return( k )\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Frame Element Member Stiffness Matrix in Global Coordinates**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def column_stiffness_matrix_global( k , T) :\n",
    "    k_global =  np.matmul( np.matmul(np.transpose( T) , k ) , T )\n",
    "                          \n",
    "    return( k_global )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example- Test the functions- Beam**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "E , A  , I , x1, y1 , x2 , y2 = 21E7 , 334E-5 , 214E-7 , 0 , 0 , 3 , 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot( [x1 ,x2] , [ y1 , y2] , \"-\", marker = \"o\", ms = 10 , lw= 5) \n",
    "plt.grid()\n",
    "plt.axhline(c=\"gray\")\n",
    "plt.axvline(c=\"gray\")\n",
    "plt.box(False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 0., 0., 0., 0., 0.],\n",
       "       [0., 1., 0., 0., 0., 0.],\n",
       "       [0., 0., 1., 0., 0., 0.],\n",
       "       [0., 0., 0., 1., 0., 0.],\n",
       "       [0., 0., 0., 0., 1., 0.],\n",
       "       [0., 0., 0., 0., 0., 1.]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = transformation_matrix( x1,y1,x2,y2)\n",
    "T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 233800.        ,       0.        ,       0.        ,\n",
       "        -233800.        ,       0.        ,       0.        ],\n",
       "       [      0.        ,    1997.33333333,    2996.        ,\n",
       "              0.        ,   -1997.33333333,    2996.        ],\n",
       "       [      0.        ,    2996.        ,    5992.        ,\n",
       "              0.        ,   -2996.        ,    2996.        ],\n",
       "       [-233800.        ,       0.        ,       0.        ,\n",
       "         233800.        ,       0.        ,       0.        ],\n",
       "       [      0.        ,   -1997.33333333,   -2996.        ,\n",
       "              0.        ,    1997.33333333,   -2996.        ],\n",
       "       [      0.        ,    2996.        ,    2996.        ,\n",
       "              0.        ,   -2996.        ,    5992.        ]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = column_stiffness_matrix( E , A  , I , x1, y1 , x2 , y2)\n",
    "k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 233800.        ,       0.        ,       0.        ,\n",
       "        -233800.        ,       0.        ,       0.        ],\n",
       "       [      0.        ,    1997.33333333,    2996.        ,\n",
       "              0.        ,   -1997.33333333,    2996.        ],\n",
       "       [      0.        ,    2996.        ,    5992.        ,\n",
       "              0.        ,   -2996.        ,    2996.        ],\n",
       "       [-233800.        ,       0.        ,       0.        ,\n",
       "         233800.        ,       0.        ,       0.        ],\n",
       "       [      0.        ,   -1997.33333333,   -2996.        ,\n",
       "              0.        ,    1997.33333333,   -2996.        ],\n",
       "       [      0.        ,    2996.        ,    2996.        ,\n",
       "              0.        ,   -2996.        ,    5992.        ]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k_glob = column_stiffness_matrix_global( k , T)\n",
    "k_glob"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example- Test the functions- Columns**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "E , A  , I , x1, y1 , x2 , y2 = 21E7 , 334E-5 , 214E-7 , 0 , 0 , 0 , 3 \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot( [x1 ,x2] , [ y1 , y2] , \"-\", marker = \"o\", ms = 10 , lw= 5) \n",
    "plt.grid()\n",
    "plt.axhline(c=\"gray\")\n",
    "plt.axvline(c=\"gray\")\n",
    "plt.box(False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  1.,  0.,  0.,  0.,  0.],\n",
       "       [-1.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  1.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  1.,  0.],\n",
       "       [ 0.,  0.,  0., -1.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  1.]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = transformation_matrix( x1,y1,x2,y2)\n",
    "T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 233800.        ,       0.        ,       0.        ,\n",
       "        -233800.        ,       0.        ,       0.        ],\n",
       "       [      0.        ,    1997.33333333,    2996.        ,\n",
       "              0.        ,   -1997.33333333,    2996.        ],\n",
       "       [      0.        ,    2996.        ,    5992.        ,\n",
       "              0.        ,   -2996.        ,    2996.        ],\n",
       "       [-233800.        ,       0.        ,       0.        ,\n",
       "         233800.        ,       0.        ,       0.        ],\n",
       "       [      0.        ,   -1997.33333333,   -2996.        ,\n",
       "              0.        ,    1997.33333333,   -2996.        ],\n",
       "       [      0.        ,    2996.        ,    2996.        ,\n",
       "              0.        ,   -2996.        ,    5992.        ]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = column_stiffness_matrix( E , A  , I , x1, y1 , x2 , y2)\n",
    "k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   1997.33333333,       0.        ,   -2996.        ,\n",
       "          -1997.33333333,       0.        ,   -2996.        ],\n",
       "       [      0.        ,  233800.        ,       0.        ,\n",
       "              0.        , -233800.        ,       0.        ],\n",
       "       [  -2996.        ,       0.        ,    5992.        ,\n",
       "           2996.        ,       0.        ,    2996.        ],\n",
       "       [  -1997.33333333,       0.        ,    2996.        ,\n",
       "           1997.33333333,       0.        ,    2996.        ],\n",
       "       [      0.        , -233800.        ,       0.        ,\n",
       "              0.        ,  233800.        ,       0.        ],\n",
       "       [  -2996.        ,       0.        ,    2996.        ,\n",
       "           2996.        ,       0.        ,    5992.        ]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k_glob = column_stiffness_matrix_global( k , T)\n",
    "k_glob"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Incilined member**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "E , A  , I , x1, y1 , x2 , y2 = 21E7 , 334E-5 , 214E-7 , 0 , 0 , 2.5 , 2.5 \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot( [x1 ,x2] , [ y1 , y2] , \"-\", marker = \"o\", ms = 10 , lw= 5) \n",
    "plt.grid()\n",
    "plt.axhline(c=\"gray\")\n",
    "plt.axvline(c=\"gray\")\n",
    "plt.box(False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.70710678,  0.70710678,  0.        ,  0.        ,  0.        ,\n",
       "         0.        ],\n",
       "       [-0.70710678,  0.70710678,  0.        ,  0.        ,  0.        ,\n",
       "         0.        ],\n",
       "       [ 0.        ,  0.        ,  1.        ,  0.        ,  0.        ,\n",
       "         0.        ],\n",
       "       [ 0.        ,  0.        ,  0.        ,  0.70710678,  0.70710678,\n",
       "         0.        ],\n",
       "       [ 0.        ,  0.        ,  0.        , -0.70710678,  0.70710678,\n",
       "         0.        ],\n",
       "       [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ,\n",
       "         1.        ]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = transformation_matrix( x1,y1,x2,y2)\n",
    "T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 198385.8785297 ,       0.        ,       0.        ,\n",
       "        -198385.8785297 ,       0.        ,       0.        ],\n",
       "       [      0.        ,    1220.25134387,    2157.12      ,\n",
       "              0.        ,   -1220.25134387,    2157.12      ],\n",
       "       [      0.        ,    2157.12      ,    5084.38059944,\n",
       "              0.        ,   -2157.12      ,    2542.19029972],\n",
       "       [-198385.8785297 ,       0.        ,       0.        ,\n",
       "         198385.8785297 ,       0.        ,       0.        ],\n",
       "       [      0.        ,   -1220.25134387,   -2157.12      ,\n",
       "              0.        ,    1220.25134387,   -2157.12      ],\n",
       "       [      0.        ,    2157.12      ,    2542.19029972,\n",
       "              0.        ,   -2157.12      ,    5084.38059944]])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k = column_stiffness_matrix( E , A  , I , x1, y1 , x2 , y2)\n",
    "k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 99803.06493678,  98582.81359292,  -1525.31417983,\n",
       "        -99803.06493678, -98582.81359292,  -1525.31417983],\n",
       "       [ 98582.81359292,  99803.06493678,   1525.31417983,\n",
       "        -98582.81359292, -99803.06493678,   1525.31417983],\n",
       "       [ -1525.31417983,   1525.31417983,   5084.38059944,\n",
       "          1525.31417983,  -1525.31417983,   2542.19029972],\n",
       "       [-99803.06493678, -98582.81359292,   1525.31417983,\n",
       "         99803.06493678,  98582.81359292,   1525.31417983],\n",
       "       [-98582.81359292, -99803.06493678,  -1525.31417983,\n",
       "         98582.81359292,  99803.06493678,  -1525.31417983],\n",
       "       [ -1525.31417983,   1525.31417983,   2542.19029972,\n",
       "          1525.31417983,  -1525.31417983,   5084.38059944]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k_glob = column_stiffness_matrix_global( k , T)\n",
    "k_glob"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "## Frame System Stiffness matrix\n",
    "\n",
    "\n",
    "**Frame System Stiffness Matrix**\n",
    "\n",
    "Determine the system stiffness matrix\n",
    "\n",
    "<img src=\"./figures/Week07-FrameExample.png\">\n",
    "\n",
    "E= 210GPa , A = 334E-5 mm2 , I = 214E-7 mm4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "import EQE512_functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: {'E': 210000000,\n",
       "  'I': 2.14e-05,\n",
       "  'A': 0.00334,\n",
       "  'jointI': [0, 0],\n",
       "  'jointJ': [0, 3]},\n",
       " 2: {'E': 210000000,\n",
       "  'I': 2.14e-05,\n",
       "  'A': 0.00334,\n",
       "  'jointI': [0, 3],\n",
       "  'jointJ': [3, 3]}}"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Mapping and numbering\n",
    "DOF = 3\n",
    "#Nodes' names with coordinates\n",
    "joint_nodes = {1:[0,0] , 2:[0,3], 3:[3,3]}\n",
    "\n",
    "# Members' names and nodes\n",
    "member_nodes= {1:[1,2] , 2:[2,3]}\n",
    "\n",
    "# Member properties\n",
    "member_1 = {\"E\":210_000_000 ,  \"I\" : 214E-7 , \"A\": 334E-5 ,   \n",
    "                \"jointI\": joint_nodes[1] , \n",
    "                \"jointJ\": joint_nodes[2]}\n",
    "\n",
    "member_2 = { \"E\":210_000_000 ,  \"I\" : 214E-7 , \"A\": 334E-5 , \n",
    "                \"jointI\": joint_nodes[2] , \n",
    "                \"jointJ\": joint_nodes[3]}\n",
    "\n",
    "member_props = {}\n",
    "\n",
    "for member_no , member_property in zip( member_nodes.keys() , [member_1 ,  member_2] ):\n",
    "    \n",
    "    member_props[ member_no ] = member_property   \n",
    "    \n",
    "member_props"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "Transformation matrix = \n",
      "[[ 0.  1.  0.  0.  0.  0.]\n",
      " [-1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 233800.               0.               0.         -233800.\n",
      "        0.               0.        ]\n",
      " [      0.            1997.33333333    2996.               0.\n",
      "    -1997.33333333    2996.        ]\n",
      " [      0.            2996.            5992.               0.\n",
      "    -2996.            2996.        ]\n",
      " [-233800.               0.               0.          233800.\n",
      "        0.               0.        ]\n",
      " [      0.           -1997.33333333   -2996.               0.\n",
      "     1997.33333333   -2996.        ]\n",
      " [      0.            2996.            2996.               0.\n",
      "    -2996.            5992.        ]]\n",
      "==================================================================================================== \n",
      " [[   1997.33333333       0.           -2996.           -1997.33333333\n",
      "        0.           -2996.        ]\n",
      " [      0.          233800.               0.               0.\n",
      "  -233800.               0.        ]\n",
      " [  -2996.               0.            5992.            2996.\n",
      "        0.            2996.        ]\n",
      " [  -1997.33333333       0.            2996.            1997.33333333\n",
      "        0.            2996.        ]\n",
      " [      0.         -233800.               0.               0.\n",
      "   233800.               0.        ]\n",
      " [  -2996.               0.            2996.            2996.\n",
      "        0.            5992.        ]]\n",
      "2\n",
      "Transformation matrix = \n",
      "[[1. 0. 0. 0. 0. 0.]\n",
      " [0. 1. 0. 0. 0. 0.]\n",
      " [0. 0. 1. 0. 0. 0.]\n",
      " [0. 0. 0. 1. 0. 0.]\n",
      " [0. 0. 0. 0. 1. 0.]\n",
      " [0. 0. 0. 0. 0. 1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 233800.               0.               0.         -233800.\n",
      "        0.               0.        ]\n",
      " [      0.            1997.33333333    2996.               0.\n",
      "    -1997.33333333    2996.        ]\n",
      " [      0.            2996.            5992.               0.\n",
      "    -2996.            2996.        ]\n",
      " [-233800.               0.               0.          233800.\n",
      "        0.               0.        ]\n",
      " [      0.           -1997.33333333   -2996.               0.\n",
      "     1997.33333333   -2996.        ]\n",
      " [      0.            2996.            2996.               0.\n",
      "    -2996.            5992.        ]]\n",
      "==================================================================================================== \n",
      " [[ 233800.               0.               0.         -233800.\n",
      "        0.               0.        ]\n",
      " [      0.            1997.33333333    2996.               0.\n",
      "    -1997.33333333    2996.        ]\n",
      " [      0.            2996.            5992.               0.\n",
      "    -2996.            2996.        ]\n",
      " [-233800.               0.               0.          233800.\n",
      "        0.               0.        ]\n",
      " [      0.           -1997.33333333   -2996.               0.\n",
      "     1997.33333333   -2996.        ]\n",
      " [      0.            2996.            2996.               0.\n",
      "    -2996.            5992.        ]]\n",
      "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"
     ]
    }
   ],
   "source": [
    "# Data structure for member stiffness matrices\n",
    "k_members = {}\n",
    "\n",
    "for member_no , member_property in member_props.items() : \n",
    "    print( member_no)\n",
    "    k_members[member_no] =  EQE512_functions.column_stiffness_matrix_global(member_property , show_T_K= True )\n",
    "    \n",
    "    print( \"=\"*100 , \"\\n\" , k_members[member_no])\n",
    "    \n",
    "print( \"~\"*100)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   1997,       0,   -2996,   -1997,       0,   -2996,       0,\n",
       "              0,       0],\n",
       "       [      0,  233800,       0,       0, -233800,       0,       0,\n",
       "              0,       0],\n",
       "       [  -2996,       0,    5992,    2996,       0,    2996,       0,\n",
       "              0,       0],\n",
       "       [  -1997,       0,    2996,  235797,       0,    2996, -233800,\n",
       "              0,       0],\n",
       "       [      0, -233800,       0,       0,  235797,    2996,       0,\n",
       "          -1997,    2996],\n",
       "       [  -2996,       0,    2996,    2996,    2996,   11984,       0,\n",
       "          -2996,    2996],\n",
       "       [      0,       0,       0, -233800,       0,       0,  233800,\n",
       "              0,       0],\n",
       "       [      0,       0,       0,       0,   -1997,   -2996,       0,\n",
       "           1997,   -2996],\n",
       "       [      0,       0,       0,       0,    2996,    2996,       0,\n",
       "          -2996,    5992]])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = EQE512_functions.system_stiffness_matrix( k_members , joint_nodes , member_nodes , DOF = 3)\n",
    "K"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check the result \n",
    "\n",
    "<img src=\"./figures/Week07-FrameExample-Stiffness_matrix.png\">\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Mapping and numbering\n",
    "DOF = 3\n",
    "#Nodes' names with coordinates\n",
    "#joint_nodes = {1:[0,0] , 2:[0,3], 3:[3,3]}\n",
    "\n",
    "# Members' names and nodes\n",
    "member_nodes= {1:[1,2] , 2:[2,3]}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAD4CAYAAADFAawfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjMsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+AADFEAAATMklEQVR4nO3df7DddX3n8efL5DIQIGDXtgsEJ0gtPwQVhqGIlKmEsVBZ0Q5u6QrrdmrTzqDlh51OtLNj6Iy72rEgM3W7pVRlhdJihKkLysqIFNxZUQhggOBoECGCgMOPBIiQwHv/ON/QGHJzzknuOeeTe5+Pme/ce8793O/n/U0+93U/93O+3/NNVSFJatdrJl2AJGn7DGpJapxBLUmNM6glqXEGtSQ1zqCWpMYZ1JLUOINakhpnUEtS45oP6iS7J/lOkruT3JvkwknXpNklyb5JViS5P8nqJG+bdE2aPZKcm+SeLr/O25F9zJ/pokbgBeCkqno2yRTwrSRfq6pvT7owzRqXADdU1RlJdgMWTLogzQ5JjgD+CDgWeBG4Icn1VfWDYfbTRFAfefmRBwMXAGcBewPrgSuAi6pqDfBs13Sq23yDEg3sr3/vtGnH159dff0TwInAfwGoqhfp/UBJA/nGTQdPO76Aw4BvV9XzAEn+FXgv8FfD9DHxpY8jLz/yVOB79H7rLATSffwj4HtHXn7kqUnmJbkLeBy4sapum1jB2qX89e+dtt3x9R/ectjZwBPA55PcmeSyJHtOrGDtUr5x08HbHV/nnf+61wInJvl3SRYAvwMcOGw/AwV1kvO79ZV7klyVZPdhO9qWbia9gt6fmlNbfXmqe37FEV84YnFVvRVYBBzb/TkhbVc3k97u+Drodb/0aeBo4G+r6ijgOWDZWAvVLqmbSW93fJ122sKLDzpot88BNwI3AHcDm4btq29QJzkA+FPgmKo6ApgHnDlsR9O4gFcf4NamgPMBqupp4GbglBnqX7Nb3/G174Ld5y3Yber5Lf5KW0EvuKV+Bsqvv79s0aKqOrqqTgSeBIZan4bBlz7mA3skmU/vt8cjw3Y0jbPoc6Cb1m2aeum5l84GSLIHcDJw/wz1r9mt7/hauMfuU7+8914LkhzSPbUEuG/klWk26Du+gKknn9z0nwGSvB74XeCqYTvqG9RV9RPg08BDwKPAM1X19a3bJVma5PZuWzpg/3v3a7DpmU386FM/Wpjke8B36a1RXzfg/jW39R1fAL979JsCXNmNsbcC/22kVWm2GGh8/eWFj++d5D7gfwPnVNVTw3bU96yPJK8FTgcOAp4GvpTkrKq6Yst2VXUpcOmQ/a+nt/A+rd0P3J1f+8tfW7fqA6vePOS+pb7jC+CA1+7zbFUdM4Z6NLsMNL4+c8n+65actObwnelokKWPk4EfVdUTVbURuAY4fmc63cIVwMY+bTYCX5yh/jS3OL40SmMbX4ME9UPAcUkWJAm9NbzVO9tx5yIGO9CLZ6g/zS2OL43S2MbXIGvUt9F7JXwlsKr7nmGXOLZp1QdWrQHOAJ7n1Qe8sXv+jK6dNJSP/PN1A42vrp00lCUnrRlofHXtdkpauAt5dz71+cDZwF70rkT8InCxIa2d1Z1Pvc3xZUhrZ3XnU29zfM1ESEMjQS1Jmt7ELyGXJG2fQS1JjTOoJalxBrUkNc6glqTGGdSS1DiDWpIaZ1BLUuMMaklqnEEtSY0zqCWpcQa1JDXOoJakxhnUktQ4g1qSGtc3qJMckuSuLbZ1Sc4bR3GSpCFvHJBkHvAT4Deq6scjq0qS9Iphlz6WAGsMaUkan2GD+kzgqlEUIknatoGDOsluwLuBL03z9aVJbu+2pTNVoCTNdQOvUSc5HTinqt452pIkSVsaZunj93HZQ5LGbqAZdZIFwMPAG6rqmZFXJUl6xVCn50mSxs8rEyWpcQa1JDXOoJakxhnUktQ4g1qSGmdQS1LjDGpJapxBLUmNM6glqXEGtSQ1zqCWpMYZ1JLUOINakhpnUEtS4wxqSWqcQS1JjTOoJalxAwV1kn2TrEhyf5LVSd426sIkST3zB2x3CXBDVZ2RZDdgwQhrkiRtoe89E5MsBO6md2Nbb7AoSWM2yNLHG4AngM8nuTPJZUn23LpRkqVJbu+2pTNeqSTNUYPMqI8Bvg28vapuS3IJsK6q/us4CpSkuW6QGfVaYG1V3dY9XgEcPbqSJElb6hvUVfVT4OEkh3RPLQHuG2lVkqRX9F36AEjyVuAyYDfgAeAPquqpEdcmSWLAoJYkTY5XJkpS4wxqSWqcQS1JjTOoJalxBrUkNc6glqTGGdSS1DiDWpIaZ1BLUuMMaklqnEEtSY0zqCWpcQa1JDXOoJakxhnUktQ4g1qSGmdQS1Lj5g/SKMmDwHrgJWBTVR0zyqIkSf9moKDuvKOqfjaySiRJ2+TShyQ1btCgLuDrSe5IsnRbDZIsTXJ7t22zjSRpeAPdhTzJ/lX1SJJfAW4EPlxVt4y8OknSYDPqqnqk+/g4cC1w7CiLkiT9m75BnWTPJHtv/hx4J3DPqAuTJPUMctbHrwLXJtnc/h+r6oaRViVJesVAa9SSpMnx9DxJapxBLUmNM6glqXEGtSQ1zqCWpMYZ1JLUOINakhpnUEtS4wxqSWqcQS1JjTOoJalxBrUkNc6glqTGNR/USQ5M8s0kq5Pcm+TcSdek2SXJg0lWJbkrye2TrkezS5Lzu+y6J8lVSXYfeh+tv81pkv2A/apqZXcDgzuA91TVfRMuTbNEkgeBY6rqZ5OuRbNLkgOAbwGHV9WGJFcDX62qLwyzn0FuHDB6y/c5GLgAOAvYG1gPXAFcVFVrgEcBqmp9ktXAAYBBrYF89k9umnZ8nfM/T1ozydq061u+fPm04wv4Ob2c3SPJRmAB8MiwfUx+Rr18n1OBFcBUt222sdvOYPkzXwNIshi4BTiiqtaNt1Dtij77Jzf1HV8f+rsl/wN4Cijg76rq0rEXql3S8uXL+46vCy+88NeBTwAbgK9X1fuH7WfgNeok85LcmeS6YTuZVm8mvYLeb5mprb461T2/guX7HJxkL+DLwHmGtAbRzaT7jq/3HvfH/6mqjgZOBc5JcuJ4K9WuqJtJb3d8bdiwYcXU1NSZwEHA/sCeSc4atq9hXkw8F1g9bAd9XMCrD3BrUxs21kfohfSVVXXNDNeg2Wug8bXkLf/x/QBV9ThwLXDsqAvTrNB3fD3wwAO7LVq0aJ+qeqKqNgLXAMcP29FAQZ1kEfAu4LJhO+jjLPocaFVNffArGz4IrK6qi2a4f81ufcfXCxs3TP38xefOBkiyJ/BO4J4x1KZdX9/xtc8++8xfv379IUkWpHeH8CXswIR30Bn1Z4A/B16erkGSpUlu77alA+53734N/u/DL/GP92yaAk7qTp+6K8nvDLh/zW19x9f6DU9x0b+ctzDJ3cB3gOur6obRl6ZZoO/4WrRoEYcffniAlcAqepk79GsgfYM6yWnA41V1x/baVdWlVXVMtw1ayPp+DU54/Xzq4wvXVdWbq+qt3fbVAfevua3v+Hrdwv352Pv+fl1VvaWq3lRVnxhHYZoV+o4vgHe84x3rq+rQqjqiqs6uqheG7WiQGfXbgXd355r+E72Z7RXDdjSNK+i9Mro9G4EvzlB/mlscXxqlsY2vvkFdVR+tqkVVtRg4E7ipqoZ+1XIaFzHYgV48Q/1pbnF8aZTGNr4mewn58mfWAGcAz/PqA97YPX9G104aSncxS9/x5UUv2hHLly8faHx17XbK5C94gc3nU58PnA3sBTxL78+Fiw1p7azufOptji9DWjurO596m+NrJkIaWglqSdK0mn/3PEma6wxqSWqcQS1JjTOoJalxBrUkNc6glqTGGdSS1DiDWpIaZ1BLUuMMaklqnEEtSY0zqCWpcQa1JDXOoJakxhnUktQ4g1qSGjfIXch3T/KdJHcnuTfJheMoTJLU0/cOL0kC7FlVzyaZAr4FnFtV3x5HgZI0183v16B6Sf5s93Cq27x/lySNyUBr1EnmJbkLeBy4sapu20abpUlu77alM12oJM1VQ93cNsm+wLXAh6vqnpFVJUl6xVBnfVTV08DNwCkjqUaS9CqDnPXxy91MmiR7ACcD94+6MElST98XE4H9gMuTzKMX7FdX1XWjLUuStNlQa9SSpPHzykRJapxBLUmNM6glqXEGtSQ1zqCWpMYZ1JLUOINakhpnUEtS4wxqSWqcQS1JjTOoJalxBrUkNc6glqTGGdSS1DiDWpIaZ1BLUuMMaklq3CD3TDwwyTeTrE5yb5Jzx1GYJKmn7624kuwH7FdVK5PsDdwBvKeq7htHgZI01/WdUVfVo1W1svt8PbAaOGDUhUmSeoZao06yGDgKuG0bX1ua5PZuWzoz5UmSBr4LeZK9gH8FPlFV14y0KknSKwaaUSeZAr4MXGlIS9J4DfJiYoDLgSer6ryxVCVJesUgQX0CcCuwCni5e/pjVfXVEdcmSWKINWpJ0mR4ZaIkNc6glqTGGdSS1DiDWpIaZ1BLUuMMaklqnEEtSY0zqCWpcQa1JDXOoJakxhnUktQ4g1qSGmdQS1LjDGpJapxBLUmNM6glqXEGtSQ1rm9QJ/lckseT3DOOgiRJv2iQGfUXgFNGXIckaRp9g7qqbgGeHEMtkqRtmLE16iRLk9zebUtnar+SNNcNdBfyJIuB66rqiFEXJEn6RZ71IUmNM6glqXGDnJ53FfD/gEOSrE3yh6MvS5K02UBr1JKkyXHpQ5IaZ1BLUuMMaklqnEEtSY0zqCWpcQa1JDXOoJakxhnUktQ4g1qSGmdQS1LjDGpJapxBLUmNM6glqXHNB7V3QdeoJZmX5M4k1026Fs0uSQ5JctcW27ok5w29n9bf5jTJicCzwP/yVmAahSQXAMcAC6vqtEnXo9kpyTzgJ8BvVNWPh/ne+aMpaTiLl11/MHABcBawN7AeuAK4qKpu6e7ZKO2Q1YceNu34Ovz7978AvAv4RNdGGsraZbdOO74WffI312zRdAmwZtiQhgZm1IuXXX8qsAKY6rbNNnbbGT/+1Gmr8ea62gGrDz1su+Pr5DU/XPnIpk3n0fsB+zNn1BrG2mW39s2vRZ/8za9BbxkXWFlVfzNsPwOtUSc5Jcn3k/wwybJhO5lON5NeASzgFw+S7vECYMU+x5/5+pnqU3NHN5Oednzd/OyzC07Yc6/j7jvk0KfHX512dd1Mum9+rV1268FJdgPeDXxpR/oa5J6J84DPAqcChwO/n+TwHelsGy7g1Qe4tak93nic92nUjtju+Fq54Xlufu7Z+cf/4Ad3Av8EnJTkirFVp13dQPkFnE8vP1dW1WM70lHfpY8kbwOWV9Vvd48/ClBV/32671m87PqbB+z/BGBev0abnn7spcf++S9eOOCPL/vugPuVeNPPHjjhNVTf8fUyeem7Tz1+z7rvXHvgv3//p1aNozbt+t7MvBNeM0B+vQwv3Xj1Xzy5x+Kjnlx47Ht/Ol27Bz/5rt+a7muDBPUZwClV9cHu8dn0XrX80FbtlgJLu4eXVtWl/Q5g8bLrXwayvTZPfOWv+PlDq3j5uac2AY8BH6+qf+i3711JkqWD/HvNJuM45tWHHtZ3fHVePvz79y9hhGvU/h/PPmuX3TrQ+Nqw8ecv//pF73waeENVPbMjfQ0S1O8DfnuroD62qj68Ix32LSi5vaqOGcW+W+Uxz35z7XjBY55Jg7yYuBY4cIvHi4BHZroQSdK2DRLU3wXemOSg7pXLM4GvjLYsSdJmfS94qapNST4E/B96C+efq6p7R1jTrF3T2g6Pefaba8cLHvOMmfgFL5Kk7Wv+TZkkaa4zqCWpcc0E9aguU29VkgOTfDPJ6iT3Jjl30jWNy1x7W9Ek+yZZkeT+7v/7bZOuadSSnN+N63uSXJVk90nXNNO29RbMSX4pyY1JftB9fO1M9NVEUI/4MvVWbQI+UlWHAccB58yBY97sXGD1pIsYo0uAG6rqUOAtzPJjT3IA8KfAMd0bqc2jd7bYbPMF4JStnlsGfKOq3gh8o3u805oIauBY4IdV9UBVvUjvfRdOn3BNI1VVj1bVyu7z9fR+eA+YbFWjl2QRvbcVvWzStYxDkoXAicA/AFTVi1U1F94Eaj6wR5L59N6caNZde1FVtwBPbvX06cDl3eeXA++Zib5aCeoDgIe3eLyWORBam3Xvt30UcNtkKxmLzwB/Drw86ULG5A3AE8Dnu+Wey5LsOemiRqmqfgJ8GngIeBR4pqq+PtmqxuZXq+pR6E3GgF+ZiZ22EtTbul5+Tpw3mGQv4MvAeVW1btL1jFKS04DHq+qOSdcyRvOBo4G/raqjgOeYoT+HW9Wty54OHATsD+yZ5KzJVrVrayWo5+Rl6kmm6IX0lVV1zaTrGYO3A+9O8iBz521F1wJrq2rzX0sr6AX3bHYy8KOqeqKqNgLXAMdPuKZxeSzJfgDdx8dnYqetBPWcu0w9SeitW66uqosmXc84VNVHq2pRVS2m9398U1XN6plWVf0UeDjJId1TS4D7JljSODwEHJdkQTfOlzDLX0DdwleAD3SffwD4l5nYaRP3TJzAZeoteDtwNrAqyV3dcx+rqq9OsCaNxoeBK7tJyAPAH0y4npGqqtuSrABW0ju76U5m4eXkSa4Cfgt4XZK1wMeBTwJXJ/lDer+w3jcjfXkJuSS1rZWlD0nSNAxqSWqcQS1JjTOoJalxBrUkNc6glqTGGdSS1Lj/DxXr0qz+W1iOAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "joint_nodes = {}\n",
    "isim = 1\n",
    "for i in range(0,13,5) :\n",
    "    for j in range(0,9,4):\n",
    "        plt.plot( i , j , marker = \"o\", ms = 10 )\n",
    "        joint_nodes[isim ] = [i,j]\n",
    "        plt.text( i+.1, j , isim)\n",
    "        isim += 1\n",
    "\n",
    "plt.axhline()\n",
    "plt.ylim(bottom = 0 )\n",
    "plt.box(False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: [0, 0],\n",
       " 2: [0, 4],\n",
       " 3: [0, 8],\n",
       " 4: [5, 0],\n",
       " 5: [5, 4],\n",
       " 6: [5, 8],\n",
       " 7: [10, 0],\n",
       " 8: [10, 4],\n",
       " 9: [10, 8]}"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "joint_nodes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: [1, 2],\n",
       " 2: [2, 3],\n",
       " 3: [4, 5],\n",
       " 4: [5, 6],\n",
       " 5: [7, 8],\n",
       " 6: [8, 9],\n",
       " 7: [2, 5],\n",
       " 8: [5, 8],\n",
       " 9: [3, 6],\n",
       " 10: [6, 9]}"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "member_nodes= {1:[1,2] , 2:[2,3] , 3:[4,5], 4:[5,6],5:[7,8] , 6:[8,9], 7:[2,5] , 8:[5,8],9:[3,6] , 10:[6,9]}\n",
    "member_nodes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Member properties\n",
    "column = {\"E\":30_000_000 ,  \"I\" : 200 , \"A\": 100 }\n",
    "beam = { \"E\":30_000_000 ,  \"I\" : 100 , \"A\": 50 }\n",
    "\n",
    "plt.figure()\n",
    "member_props = {}\n",
    "for member , nodes in member_nodes.items():\n",
    "    #print( member , joint_nodes[nodes[0]],  joint_nodes[nodes[1]])\n",
    "    \n",
    "    plt.plot( [joint_nodes[nodes[0]][0] , joint_nodes[nodes[1]][0]] ,\n",
    "            [joint_nodes[nodes[0]][1] , joint_nodes[nodes[1]][1]],c= \"k\", marker = \"o\",ms = 10) \n",
    "    \n",
    "    plt.text( joint_nodes[nodes[0]][0]+.2,joint_nodes[nodes[0]][1]+.3 , nodes[0])\n",
    "    plt.text( joint_nodes[nodes[1]][0]+.2,joint_nodes[nodes[1]][1]+.3 , nodes[1])\n",
    "    \n",
    "    \n",
    "    if joint_nodes[nodes[0]][0] ==  joint_nodes[nodes[1]][0]:\n",
    "     #   print( \"Column\")\n",
    "        member_props[member] = {\"E\":30_000_000 ,  \"I\" : 200 , \"A\": 100 , 'jointI': joint_nodes[nodes[0]],  'jointJ':joint_nodes[nodes[1]]}\n",
    "    else: \n",
    "      #  print(\"Beam\")\n",
    "        member_props[member] = {\"E\":30_000_000 ,  \"I\" : 100 , \"A\": 50, 'jointI':joint_nodes[nodes[0]],  'jointJ':joint_nodes[nodes[1]]}\n",
    "plt.axhline(c = \"black\",lw= 10)\n",
    "plt.ylim(bottom = 0 )\n",
    "plt.box(False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "Transformation matrix = \n",
      "[[ 0.  1.  0.  0.  0.  0.]\n",
      " [-1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00  1.125e+09  2.250e+09  0.000e+00 -1.125e+09  2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  6.000e+09  0.000e+00 -2.250e+09  3.000e+09]\n",
      " [-7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00 -1.125e+09 -2.250e+09  0.000e+00  1.125e+09 -2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  3.000e+09  0.000e+00 -2.250e+09  6.000e+09]]\n",
      "==================================================================================================== \n",
      " [[ 1.125e+09  0.000e+00 -2.250e+09 -1.125e+09  0.000e+00 -2.250e+09]\n",
      " [ 0.000e+00  7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  6.000e+09  2.250e+09  0.000e+00  3.000e+09]\n",
      " [-1.125e+09  0.000e+00  2.250e+09  1.125e+09  0.000e+00  2.250e+09]\n",
      " [ 0.000e+00 -7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  3.000e+09  2.250e+09  0.000e+00  6.000e+09]]\n",
      "2\n",
      "Transformation matrix = \n",
      "[[ 0.  1.  0.  0.  0.  0.]\n",
      " [-1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00  1.125e+09  2.250e+09  0.000e+00 -1.125e+09  2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  6.000e+09  0.000e+00 -2.250e+09  3.000e+09]\n",
      " [-7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00 -1.125e+09 -2.250e+09  0.000e+00  1.125e+09 -2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  3.000e+09  0.000e+00 -2.250e+09  6.000e+09]]\n",
      "==================================================================================================== \n",
      " [[ 1.125e+09  0.000e+00 -2.250e+09 -1.125e+09  0.000e+00 -2.250e+09]\n",
      " [ 0.000e+00  7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  6.000e+09  2.250e+09  0.000e+00  3.000e+09]\n",
      " [-1.125e+09  0.000e+00  2.250e+09  1.125e+09  0.000e+00  2.250e+09]\n",
      " [ 0.000e+00 -7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  3.000e+09  2.250e+09  0.000e+00  6.000e+09]]\n",
      "3\n",
      "Transformation matrix = \n",
      "[[ 0.  1.  0.  0.  0.  0.]\n",
      " [-1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00  1.125e+09  2.250e+09  0.000e+00 -1.125e+09  2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  6.000e+09  0.000e+00 -2.250e+09  3.000e+09]\n",
      " [-7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00 -1.125e+09 -2.250e+09  0.000e+00  1.125e+09 -2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  3.000e+09  0.000e+00 -2.250e+09  6.000e+09]]\n",
      "==================================================================================================== \n",
      " [[ 1.125e+09  0.000e+00 -2.250e+09 -1.125e+09  0.000e+00 -2.250e+09]\n",
      " [ 0.000e+00  7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  6.000e+09  2.250e+09  0.000e+00  3.000e+09]\n",
      " [-1.125e+09  0.000e+00  2.250e+09  1.125e+09  0.000e+00  2.250e+09]\n",
      " [ 0.000e+00 -7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  3.000e+09  2.250e+09  0.000e+00  6.000e+09]]\n",
      "4\n",
      "Transformation matrix = \n",
      "[[ 0.  1.  0.  0.  0.  0.]\n",
      " [-1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00  1.125e+09  2.250e+09  0.000e+00 -1.125e+09  2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  6.000e+09  0.000e+00 -2.250e+09  3.000e+09]\n",
      " [-7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00 -1.125e+09 -2.250e+09  0.000e+00  1.125e+09 -2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  3.000e+09  0.000e+00 -2.250e+09  6.000e+09]]\n",
      "==================================================================================================== \n",
      " [[ 1.125e+09  0.000e+00 -2.250e+09 -1.125e+09  0.000e+00 -2.250e+09]\n",
      " [ 0.000e+00  7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  6.000e+09  2.250e+09  0.000e+00  3.000e+09]\n",
      " [-1.125e+09  0.000e+00  2.250e+09  1.125e+09  0.000e+00  2.250e+09]\n",
      " [ 0.000e+00 -7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  3.000e+09  2.250e+09  0.000e+00  6.000e+09]]\n",
      "5\n",
      "Transformation matrix = \n",
      "[[ 0.  1.  0.  0.  0.  0.]\n",
      " [-1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00  1.125e+09  2.250e+09  0.000e+00 -1.125e+09  2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  6.000e+09  0.000e+00 -2.250e+09  3.000e+09]\n",
      " [-7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00 -1.125e+09 -2.250e+09  0.000e+00  1.125e+09 -2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  3.000e+09  0.000e+00 -2.250e+09  6.000e+09]]\n",
      "==================================================================================================== \n",
      " [[ 1.125e+09  0.000e+00 -2.250e+09 -1.125e+09  0.000e+00 -2.250e+09]\n",
      " [ 0.000e+00  7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  6.000e+09  2.250e+09  0.000e+00  3.000e+09]\n",
      " [-1.125e+09  0.000e+00  2.250e+09  1.125e+09  0.000e+00  2.250e+09]\n",
      " [ 0.000e+00 -7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  3.000e+09  2.250e+09  0.000e+00  6.000e+09]]\n",
      "6\n",
      "Transformation matrix = \n",
      "[[ 0.  1.  0.  0.  0.  0.]\n",
      " [-1.  0.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]\n",
      " [ 0.  0.  0. -1.  0.  0.]\n",
      " [ 0.  0.  0.  0.  0.  1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00  1.125e+09  2.250e+09  0.000e+00 -1.125e+09  2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  6.000e+09  0.000e+00 -2.250e+09  3.000e+09]\n",
      " [-7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00  0.000e+00]\n",
      " [ 0.000e+00 -1.125e+09 -2.250e+09  0.000e+00  1.125e+09 -2.250e+09]\n",
      " [ 0.000e+00  2.250e+09  3.000e+09  0.000e+00 -2.250e+09  6.000e+09]]\n",
      "==================================================================================================== \n",
      " [[ 1.125e+09  0.000e+00 -2.250e+09 -1.125e+09  0.000e+00 -2.250e+09]\n",
      " [ 0.000e+00  7.500e+08  0.000e+00  0.000e+00 -7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  6.000e+09  2.250e+09  0.000e+00  3.000e+09]\n",
      " [-1.125e+09  0.000e+00  2.250e+09  1.125e+09  0.000e+00  2.250e+09]\n",
      " [ 0.000e+00 -7.500e+08  0.000e+00  0.000e+00  7.500e+08  0.000e+00]\n",
      " [-2.250e+09  0.000e+00  3.000e+09  2.250e+09  0.000e+00  6.000e+09]]\n",
      "7\n",
      "Transformation matrix = \n",
      "[[1. 0. 0. 0. 0. 0.]\n",
      " [0. 1. 0. 0. 0. 0.]\n",
      " [0. 0. 1. 0. 0. 0.]\n",
      " [0. 0. 0. 1. 0. 0.]\n",
      " [0. 0. 0. 0. 1. 0.]\n",
      " [0. 0. 0. 0. 0. 1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "==================================================================================================== \n",
      " [[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "8\n",
      "Transformation matrix = \n",
      "[[1. 0. 0. 0. 0. 0.]\n",
      " [0. 1. 0. 0. 0. 0.]\n",
      " [0. 0. 1. 0. 0. 0.]\n",
      " [0. 0. 0. 1. 0. 0.]\n",
      " [0. 0. 0. 0. 1. 0.]\n",
      " [0. 0. 0. 0. 0. 1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "==================================================================================================== \n",
      " [[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "9\n",
      "Transformation matrix = \n",
      "[[1. 0. 0. 0. 0. 0.]\n",
      " [0. 1. 0. 0. 0. 0.]\n",
      " [0. 0. 1. 0. 0. 0.]\n",
      " [0. 0. 0. 1. 0. 0.]\n",
      " [0. 0. 0. 0. 1. 0.]\n",
      " [0. 0. 0. 0. 0. 1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "==================================================================================================== \n",
      " [[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "10\n",
      "Transformation matrix = \n",
      "[[1. 0. 0. 0. 0. 0.]\n",
      " [0. 1. 0. 0. 0. 0.]\n",
      " [0. 0. 1. 0. 0. 0.]\n",
      " [0. 0. 0. 1. 0. 0.]\n",
      " [0. 0. 0. 0. 1. 0.]\n",
      " [0. 0. 0. 0. 0. 1.]]\n",
      "and member stiffnes matrix = \n",
      "[[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "==================================================================================================== \n",
      " [[ 3.00e+08  0.00e+00  0.00e+00 -3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00  2.88e+08  7.20e+08  0.00e+00 -2.88e+08  7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  2.40e+09  0.00e+00 -7.20e+08  1.20e+09]\n",
      " [-3.00e+08  0.00e+00  0.00e+00  3.00e+08  0.00e+00  0.00e+00]\n",
      " [ 0.00e+00 -2.88e+08 -7.20e+08  0.00e+00  2.88e+08 -7.20e+08]\n",
      " [ 0.00e+00  7.20e+08  1.20e+09  0.00e+00 -7.20e+08  2.40e+09]]\n",
      "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"
     ]
    }
   ],
   "source": [
    "# Data structure for member stiffness matrices\n",
    "k_members = {}\n",
    "\n",
    "for member_no , member_property in member_props.items() : \n",
    "    print( member_no)\n",
    "    k_members[member_no] =  EQE512_functions.column_stiffness_matrix_global(member_property , show_T_K= True )\n",
    "    \n",
    "    print( \"=\"*100 , \"\\n\" , k_members[member_no])\n",
    "    \n",
    "print( \"~\"*100)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1125000000,           0, -2147483648, -1125000000,           0,\n",
       "        -2147483648,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,   750000000,           0,           0,  -750000000,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [-2147483648,           0, -2147483648, -2147483648,           0,\n",
       "        -2147483648,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [-1125000000,           0, -2147483648, -1847483648,           0,\n",
       "        -2147483648, -1125000000,           0, -2147483648,           0,\n",
       "                  0,           0,  -300000000,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,  -750000000,           0,           0,  1788000000,\n",
       "          720000000,           0,  -750000000,           0,           0,\n",
       "                  0,           0,           0,  -288000000,   720000000,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [-2147483648,           0, -2147483648, -2147483648,   720000000,\n",
       "          252516352, -2147483648,           0, -2147483648,           0,\n",
       "                  0,           0,           0,  -720000000,  1200000000,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0, -1125000000,           0,\n",
       "        -2147483648,  1425000000,           0, -2147483648,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "         -300000000,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,  -750000000,\n",
       "                  0,           0,  1038000000,   720000000,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,  -288000000,   720000000,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0, -2147483648,           0,\n",
       "        -2147483648, -2147483648,   720000000,   252516352,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,  -720000000,  1200000000,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,  1125000000,\n",
       "                  0, -2147483648, -1125000000,           0, -2147483648,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "          750000000,           0,           0,  -750000000,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0, -2147483648,\n",
       "                  0, -2147483648, -2147483648,           0, -2147483648,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,  -300000000,           0,\n",
       "                  0,           0,           0,           0, -1125000000,\n",
       "                  0, -2147483648, -1547483648,           0, -2147483648,\n",
       "        -1125000000,           0, -2147483648,           0,           0,\n",
       "                  0,  -300000000,           0,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,  -288000000,\n",
       "         -720000000,           0,           0,           0,           0,\n",
       "         -750000000,           0,           0,  2076000000,           0,\n",
       "                  0,  -750000000,           0,           0,           0,\n",
       "                  0,           0,  -288000000,   720000000,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,   720000000,\n",
       "         1200000000,           0,           0,           0, -2147483648,\n",
       "                  0, -2147483648, -2147483648,           0, -2147483648,\n",
       "        -2147483648,           0, -2147483648,           0,           0,\n",
       "                  0,           0,  -720000000,  1200000000,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,  -300000000,           0,           0,           0,\n",
       "                  0,           0, -1125000000,           0, -2147483648,\n",
       "         1725000000,           0, -2147483648,           0,           0,\n",
       "                  0,           0,           0,           0,  -300000000,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,  -288000000,  -720000000,           0,\n",
       "                  0,           0,           0,  -750000000,           0,\n",
       "                  0,  1326000000,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "         -288000000,   720000000],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,   720000000,  1200000000,           0,\n",
       "                  0,           0, -2147483648,           0, -2147483648,\n",
       "        -2147483648,           0, -2147483648,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "         -720000000,  1200000000],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,  1125000000,           0,\n",
       "        -2147483648, -1125000000,           0, -2147483648,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,   750000000,\n",
       "                  0,           0,  -750000000,           0,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0, -2147483648,           0,\n",
       "        -2147483648, -2147483648,           0, -2147483648,           0,\n",
       "                  0,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,  -300000000,           0,           0,\n",
       "                  0,           0,           0, -1125000000,           0,\n",
       "        -2147483648, -1847483648,           0, -2147483648, -1125000000,\n",
       "                  0, -2147483648],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,  -288000000,  -720000000,\n",
       "                  0,           0,           0,           0,  -750000000,\n",
       "                  0,           0,  1788000000,  -720000000,           0,\n",
       "         -750000000,           0],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,   720000000,  1200000000,\n",
       "                  0,           0,           0, -2147483648,           0,\n",
       "        -2147483648, -2147483648,  -720000000,   252516352, -2147483648,\n",
       "                  0, -2147483648],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "         -300000000,           0,           0,           0,           0,\n",
       "                  0, -1125000000,           0, -2147483648,  1425000000,\n",
       "                  0, -2147483648],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,  -288000000,  -720000000,           0,           0,\n",
       "                  0,           0,  -750000000,           0,           0,\n",
       "         1038000000,  -720000000],\n",
       "       [          0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,           0,           0,           0,           0,\n",
       "                  0,   720000000,  1200000000,           0,           0,\n",
       "                  0, -2147483648,           0, -2147483648, -2147483648,\n",
       "         -720000000,   252516352]])"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = EQE512_functions.system_stiffness_matrix( k_members , joint_nodes , member_nodes , DOF = 3)\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variable                         Type        Data/Info\n",
      "------------------------------------------------------\n",
      "A                                float       0.00334\n",
      "DOF                              int         3\n",
      "E                                float       210000000.0\n",
      "EQE512_functions                 module      <module 'EQE512_functions<...>is\\\\EQE512_functions.py'>\n",
      "I                                float       2.14e-05\n",
      "K                                ndarray     27x27: 729 elems, type `int32`, 2916 bytes\n",
      "T                                ndarray     6x6: 36 elems, type `float64`, 288 bytes\n",
      "beam                             dict        n=3\n",
      "column                           dict        n=3\n",
      "column_stiffness_matrix          function    <function column_stiffnes<...>ix at 0x000001F122002550>\n",
      "column_stiffness_matrix_global   function    <function column_stiffnes<...>al at 0x000001F1220025E0>\n",
      "i                                int         10\n",
      "isim                             int         10\n",
      "j                                int         8\n",
      "joint_nodes                      dict        n=9\n",
      "k                                ndarray     6x6: 36 elems, type `float64`, 288 bytes\n",
      "k_glob                           ndarray     6x6: 36 elems, type `float64`, 288 bytes\n",
      "k_members                        dict        n=10\n",
      "member                           int         10\n",
      "member_1                         dict        n=5\n",
      "member_2                         dict        n=5\n",
      "member_no                        int         10\n",
      "member_nodes                     dict        n=10\n",
      "member_property                  dict        n=5\n",
      "member_props                     dict        n=10\n",
      "nodes                            list        n=2\n",
      "np                               module      <module 'numpy' from 'C:\\<...>ges\\\\numpy\\\\__init__.py'>\n",
      "plt                              module      <module 'matplotlib.pyplo<...>\\\\matplotlib\\\\pyplot.py'>\n",
      "transformation_matrix            function    <function transformation_<...>ix at 0x000001F121BC3D30>\n",
      "x1                               int         0\n",
      "x2                               float       2.5\n",
      "y1                               int         0\n",
      "y2                               float       2.5\n"
     ]
    }
   ],
   "source": [
    "whos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## Support settlements"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "---\n",
    "## Elastic supports"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ {\\large\n",
    "K = \\begin{bmatrix} \n",
    "a & 0 & 0 & a & 0 & 0 \\\\\n",
    "0 & b & c & 0 & 0 & 0 \\\\\n",
    "0 & c & d & 0 & 0 & 0 \\\\\n",
    "a & 0 & 0 & a & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & b & c \\\\\n",
    "0 & 0 & 0 & 0 & b & c\n",
    "\\end{bmatrix}}$$\n",
    "\n",
    "\n",
    "K_axial_stifness = e + K\n",
    "\n",
    "\n",
    "$$ {\\large\n",
    "K = \\begin{bmatrix} \n",
    "a+e & 0 & 0 & a+e & 0 & 0 \\\\\n",
    "0 & b & c & 0 & 0 & 0 \\\\\n",
    "0 & c & d & 0 & 0 & 0 \\\\\n",
    "a+e & 0 & 0 & a+e & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & b & c \\\\\n",
    "0 & 0 & 0 & 0 & b & c\n",
    "\\end{bmatrix}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**NEXT WEEK**\n",
    "The topics for the next week \n",
    "\n",
    "- \"Midterm Exam\"\n",
    "\n",
    "- Term Project : Please set your groups in GTUZEM: Please select your groups until 2021_1115 8am using the [GTUZEM Moodle site](https://gtuzem.gtu.edu.tr/UE2/mod/choice/view.php?id=19200)"
   ]
  }
 ],
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