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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 06: Matrix methods"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.4 Pg.No.205"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import math\n",
      "import numpy as np\n",
      "from sympy import solve, symbols, pprint\n",
      "from sympy import diff\n",
      "x,y=symbols('x,y')\n",
      "\n",
      "E=200000                #youngs modulus (N/mm^2)\n",
      "nu=0.3                   #poissons ratio\n",
      "u1=0.001\n",
      "u2=0.003\n",
      "u3=-0.003              #displacements of corners (m)\n",
      "u4=0\n",
      "v1=-0.004\n",
      "v2=-0.002\n",
      "v3=0.001\n",
      "v4=0.001\n",
      "\n",
      "\n",
      "a=np.array([[1,-2,-1,2],[1,2,-1,-2],[1,2,1,2],[1,-2,1,-2]])\n",
      "b=np.array([u1,u2,u3,u4])\n",
      "alpha=np.linalg.solve(a,b)\n",
      "alpha1=alpha[0]\n",
      "alpha2=alpha[1]\n",
      "alpha3=alpha[2]\n",
      "alpha4=alpha[3]\n",
      "\n",
      "a=np.array([[1,-2,-1,2],[1,2,-1,-2],[1,2,1,2],[1,-2,1,-2]])\n",
      "b=np.array([v1,v2,v3,v4])\n",
      "alpha=np.linalg.solve(a,b)\n",
      "alpha5=alpha[0]\n",
      "alpha6=alpha[1]\n",
      "alpha7=alpha[2]\n",
      "alpha8=alpha[3]\n",
      "\n",
      "u=alpha1+alpha2*x+alpha3*y+alpha4*x*y\n",
      "v=alpha5+alpha6*x+alpha7*y+alpha8*x*y\n",
      "\n",
      "ex=diff(u,x)\n",
      "ey=diff(v,y)\n",
      "Yxy=diff(u,y)+diff(v,x)\n",
      "\n",
      "ex=-0.000125\n",
      "ey=.002\n",
      "Yxy=-0.0015\n",
      "\n",
      "sigma_x=E/(1-nu**2)*(ex+nu*ey)\n",
      "sigma_y=E/(1-nu**2)*(ey+nu*ex)\n",
      "print \"longitudinal stress in x direction = %3.1f N/mm^2\"%(sigma_x)\n",
      "print \"longitudinal stress in y direction = %3.1f N/mm^2\"%(sigma_y)\n",
      "\n",
      "T_xy=E/2/(1+nu)*Yxy\n",
      "print \"shear stress at the center of plate = %3.1f N/mm^2\"%(T_xy)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "longitudinal stress in x direction = 104.4 N/mm^2\n",
        "longitudinal stress in y direction = 431.3 N/mm^2\n",
        "shear stress at the center of plate = -115.4 N/mm^2\n"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
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