{
 "metadata": {
  "name": "chapter 8.ipynb"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter No.8:Thin And Thick Cyclinders And Spheres"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.1,Page No.322"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "L=3000 #mm #Length\n",
      "d1=1000 #mm #Internal diameter\n",
      "t=15 #mm #Thickness\n",
      "P=1.5 #N/mm**2 #Fluid Pressure\n",
      "E=2*10**5 #n/mm**2 #Modulus of elasticity\n",
      "mu=0.3 #Poissoin's ratio\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Hoop stress\n",
      "f1=P*d1*(2*t)**-1 #N/mm**2\n",
      "\n",
      "#Longitudinal Stress\n",
      "f2=P*d1*(4*t)**-1 #N/mm**2\n",
      "\n",
      "#Max shear stress\n",
      "q_max=(f1-f2)*2**-1 #N/mm**2\n",
      "\n",
      "#Diametrical Strain\n",
      "#Let e1=dell_d*d**-1  .....................(1)\n",
      "e1=(f1-mu*f2)*E**-1 \n",
      "\n",
      "#Sub values in equation 1 and further simplifying we get\n",
      "dell_d=e1*d1 #mm\n",
      "\n",
      "#Longitudinal strain\n",
      "#e2=dell_L*L**-1      ......................(2)\n",
      "e2=(f2-mu*f1)*E**-1 \n",
      "\n",
      "#Sub values in equation 2 and further simplifying we get\n",
      "dell_L=e2*L #mm\n",
      "\n",
      "#Change in Volume \n",
      "#Let Z=dell_V*V**-1      ................(3)\n",
      "Z=2*e1+e2\n",
      "\n",
      "#Sub values in equation 3 and further simplifying we get\n",
      "dell_V=Z*pi*4**-1*d1**2*L\n",
      "\n",
      "#Result\n",
      "print\"Max Intensity of shear stress\",round(q_max,2),\"N/mm**2\"\n",
      "print\"Change in the Dimensions of the shell is:dell_d\",round(dell_d,2),\"mm\"\n",
      "print\"                                        :dell_L\",round(dell_L,2),\"mm\"\n",
      "print\"                                        :dell_V\",round(dell_V,2),\"mm**3\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Max Intensity of shear stress 12.5 N/mm**2\n",
        "Change in the Dimensions of the shell is:dell_d 0.21 mm\n",
        "                                        :dell_L 0.15 mm\n",
        "                                        :dell_V 1119192.38 mm**3\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.2,Page No.323"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "L=2000 #mm #Length\n",
      "d=200 #mm # diameter\n",
      "t=10 #mm #Thickness\n",
      "dell_V=25000 #mm**3 #Additional volume\n",
      "E=2*10**5 #n/mm**2 #Modulus of elasticity\n",
      "mu=0.3 #Poissoin's ratio\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Let p be the pressure developed\n",
      "\n",
      "#Circumferential Stress\n",
      "\n",
      "#f1=p*d*(2*t)**-1 #N/mm**2\n",
      "#After sub values and further simplifying\n",
      "#f1=10*p\n",
      "\n",
      "#f1=p*d*(4*t)**-1 #N/mm**2\n",
      "#After sub values and further simplifying\n",
      "#f1=5*p\n",
      "\n",
      "#Diameterical strain = Circumferential stress\n",
      "#Let X=dell_d*d**-1     ................................(1)\n",
      "#X=e1=(f1-mu*f2)*E**-1 \n",
      "#After sub values and further simplifying\n",
      "#e1=8.5*p*E**-1\n",
      "\n",
      "#Longitudinal strain\n",
      "#Let Y=dell_L*L**-1    ......................................(2)\n",
      "#Y=e2=(f2-mu*f1)*E**-1 \n",
      "#After sub values and further simplifying\n",
      "#e2=2*p*E**-1\n",
      "\n",
      "#Volumetric strain\n",
      "#Let X=dell_V*V**-1    \n",
      "#X=2*e1+e2\n",
      "#After sub values and further simplifying\n",
      "#X=19*p*E**-1\n",
      "#After further simplifying we get\n",
      "p=dell_V*(pi*4**-1*d**2*L)**-1*E*19**-1 #N/mm**2\n",
      "\n",
      "#Hoop Stress\n",
      "f1=p*d*(2*t)**-1\n",
      "\n",
      "X=e1=8.5*p*E**-1\n",
      "#Sub value of X in equation 1 we get\n",
      "dell_d=8.5*p*E**-1*d\n",
      "\n",
      "Y=e2=2*p*E**-1\n",
      "#Sub value of Y in equation 2 we get\n",
      "dell_L=2*p*E**-1*L\n",
      "\n",
      "#Result\n",
      "print\"Pressure Developed is\",round(p,2),\"N/mm**2\"\n",
      "print\"Hoop stress Developed is\",round(f1,2),\"N/mm**2\"\n",
      "print\"Change in diameter is\",round(dell_d,2),\"mm\"\n",
      "print\"Change in Length is\",round(dell_L,2),\"mm\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Pressure Developed is 4.19 N/mm**2\n",
        "Hoop stress Developed is 41.88 N/mm**2\n",
        "Change in diameter is 0.04 mm\n",
        "Change in Length is 0.08 mm\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.3,Page No.324"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d=750 #mm #Diameter of water supply pipes\n",
      "h=50*10**3 #mm #Water head\n",
      "sigma=20 #N/mm**2 #Permissible stress\n",
      "rho=9810*10**-9 #N/mm**3\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Pressure of water\n",
      "P=rho*h #N/mm**2\n",
      "\n",
      "#Stress\n",
      "#sigma=p*d*(2*t)**-1\n",
      "#After further simplifying\n",
      "t=P*d*(2*sigma)**-1 #mm \n",
      "\n",
      "#Result\n",
      "print\"Thickness of seamless pipe is\",round(t,3),\"mm\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Thickness of seamless pipe is 9.197 mm\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.4,Page No.326"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d=2500 #mm #Diameter of riveted boiler\n",
      "P=1 #N/mm**2 #Pressure\n",
      "rho1=0.7 #Percent efficiency\n",
      "rho2=0.4 #Circumferential joints\n",
      "sigma=150 #N/mm**2 #Permissible stress\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Equating Bursting force to longitudinal joint strength ,we get\n",
      "#p*d*L=rho1*2*t*L*sigma\n",
      "#After rearranging and further simplifying we get\n",
      "t=P*d*(2*sigma*rho1)**-1 #mm\n",
      "\n",
      "#Considering Longitudinal force\n",
      "#pi*d**2*4**-1*P=rho2*pi*d*t*sigma\n",
      "#After rearranging and further simplifying we get\n",
      "t2=P*d*(4*sigma*rho2)**-1\n",
      "\n",
      "#Result\n",
      "print\"Thickness of plate required is\",round(t,2),\"mm\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Thickness of plate required is 11.9 mm\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.5,Page No.326"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "#Boiler Dimensions\n",
      "t=16 #mm #Thickness\n",
      "p=2 #N/mm**2 #internal pressure\n",
      "f=150 #N/mm**2 #Permissible stress\n",
      "rho1=0.75 #Longitudinal joints\n",
      "rho2=0.45 #circumferential joints\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Equating Bursting force to longitudinal joint strength ,we get\n",
      "d1=rho1*2*t*f*p**-1 #mm\n",
      "\n",
      "#Considering circumferential strength \n",
      "d2=4*rho2*t*f*p**-1 #mm\n",
      "\n",
      "#Result\n",
      "print\"Largest diameter of Boiler is\",round(d1,2),\"mm\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Largest diameter of Boiler is 1800.0 mm\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.6,Page No.329"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d=250 #mm #Diameter iron pipe\n",
      "t=10 #mm #Thickness\n",
      "d2=6 #mm #Diameter of steel\n",
      "p=80 #N/mm**2 #stress\n",
      "P=3 #N/mm**2 #Pressure\n",
      "E_c=1*10**5 #N/mm**2\n",
      "mu=0.3 #poissoin's ratio\n",
      "E_s=2*10**5 #N/mm**2\n",
      "n=1 #No.of wires\n",
      "\n",
      "#Calculations\n",
      "\n",
      "L=6 #mm #Length of cyclinder\n",
      "\n",
      "#Force Exerted by steel wire at diameterical section\n",
      "F=p*2*pi*d2**2*1*4**-1 #N\n",
      "\n",
      "#Initial stress in cyclinder\n",
      "f_c=F*(2*t*d2)**-1 #N/mm**2\n",
      "\n",
      "#LEt due to fluid pressure alone stresses developed in steel wire be F_w and in cyclinder f1 and f2\n",
      "f2=P*d*(4*t)**-1 #N/mm**2\n",
      "\n",
      "#Considering the equilibrium of half the cyclinder, 6mm long we get\n",
      "#F_w*2*pi*4**-1*d2**2*n+f1*2*t*d2=P*d*d2\n",
      "#After further simplifying we get\n",
      "#F_w+2.122*f1=79.58        . ......................................(1)\n",
      "\n",
      "#Equating strain in wire to circumferential strain in cyclinder \n",
      "#F_w=(f1-mu*f2)*E_s*E_c**-1 #N/mm**2\n",
      "#After further simplifying we get\n",
      "#F_w=2*f1-11.25    ....................................(2)\n",
      "\n",
      "#Sub in equation in1 we get\n",
      "f1=(79.58+11.25)*(4.122)**-1 #N/mm**2\n",
      "F_w=2*f1-11.25  #N/mm**2\n",
      "\n",
      "#Final stresses\n",
      "#1) In steel Wire\n",
      "sigma=F_w+p #N/mm**2\n",
      "\n",
      "#2) In Cyclinder\n",
      "sigma2=f1-f_c\n",
      "\n",
      "#Result\n",
      "print\"Final Stresses developed in:cyclinder is\",round(sigma,2),\"N/mm**2\"\n",
      "print\"                           :Steel is\",round(sigma2,2),\"N/mm**2\"   "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Final Stresses developed in:cyclinder is 112.82 N/mm**2\n",
        "                           :Steel is -15.66 N/mm**2\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.7,Page No.332"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d=750 #mm #Diameter of shell\n",
      "t=8 #mm #THickness\n",
      "p=2.5 #N/mm**2\n",
      "E=2*10**5 #N/mm**2\n",
      "mu=0.25 #Poissoin's ratio\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Hoop stress\n",
      "f1=f2=p*d*(4*t)**-1 #N/mm**2\n",
      "\n",
      "#Change in Diameter\n",
      "dell_d=d*p*d*(1-mu)*(4*t*E)**-1 #mm\n",
      "\n",
      "#Change in Volume\n",
      "dell_V=3*p*d*(1-mu)*(4*t*E)**-1*pi*6**-1*d**3\n",
      "\n",
      "#Answer for Change in diameter is incorrect in book\n",
      "\n",
      "#Result\n",
      "print\"Stress introduced is\",round(f1,2),\"N/mm**2\"\n",
      "print\"Change in Diameter is\",round(dell_d,2),\"N/mm**2\"\n",
      "print\"Change in Volume is\",round(dell_V,2),\"mm**3\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Stress introduced is 58.59 N/mm**2\n",
        "Change in Diameter is 0.16 N/mm**2\n",
        "Change in Volume is 145608.33 mm**3\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.8,Page No.333"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d=600 #mm #Diameter of sherical shell\n",
      "t=10 #mm #Thickness\n",
      "f=80 #N/mm**2 #Permissible stress\n",
      "rho=0.75 #Efficiency joint\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Max Pressure\n",
      "p=f*4*t*rho*d**-1 #N/mm**2\n",
      "\n",
      "#Result\n",
      "print\"Max Pressure is\",round(p,2),\"N/mm**2\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Max Pressure is 4.0 N/mm**2\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.9,Page No.333"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "L=1000 #mm #Length of shell\n",
      "d=200 #mm #Diameter\n",
      "t=6 #mm #Thickness\n",
      "p=1.5 #N/mm**2 #Internal Pressure\n",
      "E=2*10**5 #N/mm**2\n",
      "mu=0.25 #Poissoin's Ratio\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Change in Volume of sphere\n",
      "dell_V_s=3*p*d*(1-mu)*(4*t*E)**-1*pi*6**-1*d**3\n",
      "\n",
      "#Hoop stress\n",
      "f1=p*d*(2*t)**-1 #N/mm**2\n",
      "\n",
      "#Longitudinal stress\n",
      "f2=p*d*(4*t)**-1 #N/mm**2\n",
      "\n",
      "#Principal strain\n",
      "e1=(f1-mu*f2)*E**-1\n",
      "e2=(f2-mu*f1)*E**-1\n",
      "\n",
      "V_c=1000 #mm**3\n",
      "\n",
      "#Change in Volume of cyclinder\n",
      "dell_V_c=(2*e1+e2)*pi*4**-1*d**2*L\n",
      "\n",
      "#Total Change in Diameter\n",
      "dell_V=dell_V_s+dell_V_c #mm**3\n",
      "\n",
      "#Result\n",
      "print\"Change in Volume is\",round(dell_V,2),\"mm**3\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Change in Volume is 8443.03 mm**3\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.10,Page No.337"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "%matplotlib  inline\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d1=400 #mm #Internal Diameter\n",
      "t=100 #mm #Thickness\n",
      "p=80 #N/mm**2 #Fluid pressure\n",
      "\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Internal Radius\n",
      "r1=d1*2**-1 #mm\n",
      "\n",
      "#Outer Radius\n",
      "r_o=r1+t #mm\n",
      "\n",
      "p1=80 #N/mm**2\n",
      "p2=0\n",
      "\n",
      "#Now From Lame's Euation\n",
      "#p_x=b*(x**2)**-1-a\n",
      "#at x=200 #mm \n",
      "p_x=80 #N/mm**2\n",
      "#80=b*(200**2)**-1-a     ..........................(1)\n",
      "\n",
      "#at x=300 #mm\n",
      "#p_x2=0\n",
      "#0=b*(300**2)**-1-a       ...........................(2)\n",
      "\n",
      "#Sub equation 2 from 1\n",
      "#80=b*(200**2)**-1-b*(300**2)**-1\n",
      "#After Further simplifying we get\n",
      "b=(50000)**-1*(200**2*300**2*80)\n",
      "\n",
      "#From equation 2 we get\n",
      "a=b*(300**2)**-1\n",
      "\n",
      "#Variation of radial pressure p_x;\n",
      "#p_x=b*(x**2)**-1-a\n",
      "#After sub values and further simplifying we get\n",
      "\n",
      "#Radial pressure Variation\n",
      "#At \n",
      "x=200 #mm\n",
      "p_x=b*(x**2)**-1-a #N/mm**2\n",
      "\n",
      "#At\n",
      "x2=250 #mm\n",
      "p_x2=b*(x2**2)**-1-a #N/mm**2\n",
      "\n",
      "#At \n",
      "x3=300 #mm\n",
      "p_x3=b*(x3**2)**-1-a #N/mm**2\n",
      "\n",
      "\n",
      "#Hoop stress Distribution\n",
      "#Variation of F_x\n",
      "\n",
      "#At \n",
      "x=200 #mm\n",
      "F_x=b*(x**2)**-1+a #N/mm**2\n",
      "\n",
      "#At\n",
      "x2=250 #mm\n",
      "F_x2=b*(x2**2)**-1+a #N/mm**2\n",
      "\n",
      "#At\n",
      "x3=300 #mm\n",
      "F_x3=b*(x3**2)**-1+a #N/mm**2\n",
      "\n",
      "#Result\n",
      "print\"Max Hoop stress is\",round(F_x,2),\"N/mm**2\"\n",
      "print\"Min Hoop stress is\",round(F_x3,2),\"N/mm**2\"\n",
      "print\"Plot of Hoop stress\"\n",
      "\n",
      "#Plotting Variation of hoop stress\n",
      "\n",
      "X1=[x,x2,x3]\n",
      "Y1=[p_x,p_x2,p_x3]\n",
      "Y2=[-F_x,-F_x2,-F_x3]\n",
      "Z1=[0,0,0]\n",
      "plt.plot(X1,Y1,X1,Y2,X1,Z1)\n",
      "plt.xlabel(\"Length x in mm\")\n",
      "plt.ylabel(\"Radial Stress Distribution & Hoop Stress Distribution in N/mm**2\")\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Max Hoop stress is 208.0 N/mm**2\n",
        "Min Hoop stress is 128.0 N/mm**2\n",
        "Plot of Hoop stress\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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psPpzP4zq6moEBwfj999/N6pCuTGRkD7V1UBaWt0zXZyc6p7p8ucyJqJ2Q9YFiQEBAdJ0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\nRERkEiYSIiIyCRMJERGZhImEiIhM8v8BOmQpuVllnG0AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4e17390>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.11,Page No.338"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "%matplotlib  inline\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d_o=300 #mm #Outside diameter \n",
      "d2=200 #mm #Internal Diameter\n",
      "p=14 #N/mm**2 #internal Fluid pressure\n",
      "t=50 #mm #Thickness\n",
      "r_o=150 #mm #Outside Diameter\n",
      "r2=100 #mm #Internal Diameter\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#From Lame's Equation\n",
      "#p_x=b*(x**2)**-1-a #N/mm**2    ...................(1)\n",
      "#F_x=b*(x**2)**-1+a #N/mm**2   ...................(2)\n",
      "\n",
      "#At \n",
      "x=r2=100 #mm\n",
      "p_x=14 #N/mm**2\n",
      "\n",
      "#Sub value of p_x in equation 1 we get\n",
      "#14=(100)**-1*b-a    ............................(3)\n",
      "\n",
      "#At\n",
      "x2=r_o=150 #mm\n",
      "p_x2=0 #N/mm**2\n",
      "\n",
      "#Sub value in equation 1 we get\n",
      "#0=b*(150**2)**-1-a   ......................(4)\n",
      "\n",
      "#From Equations 3 and 4 we get\n",
      "#14=b*(100**2)**-1-b*(100**2)**-1\n",
      "#After sub values and further simplifying we get\n",
      "b=14*100**2*150**2*(150**2-100**2)**-1\n",
      "\n",
      "#From equation 4 we get\n",
      "a=b*(150**2)**-1\n",
      "\n",
      "#Hoop Stress\n",
      "#F_x=b*(x**2)**-1+a #N/mm**2\n",
      "\n",
      "#At \n",
      "x=100 #mm\n",
      "F_x=b*(x**2)**-1+a #N/mm**2\n",
      "\n",
      "#At\n",
      "x2=125 #mm\n",
      "F_x2=b*(x2**2)**-1+a #N/mm**2\n",
      "\n",
      "#At\n",
      "x3=150 #mm\n",
      "F_x3=b*(x3**2)**-1+a #N/mm**2\n",
      "\n",
      "#If thin Cyclindrical shell theory is used,hoop stress is uniform and is given by\n",
      "F=p*d2*(2*t)**-1 #N/mm**2\n",
      "\n",
      "#Percentage error in estimating max hoop tension\n",
      "E=(F_x-F)*F_x**-1*100 #%\n",
      "\n",
      "#Result\n",
      "print\"Max Hoop Stress Developed in the cross-section is\",round(F,2),\"N/mm**2\"\n",
      "print\"Plot of Variation of hoop stress\"\n",
      "\n",
      "#Plotting Variation of hoop stress\n",
      "\n",
      "X1=[x,x2,x3]\n",
      "Y1=[F_x,F_x2,F_x3]\n",
      "Z1=[0,0,0]\n",
      "plt.plot(X1,Y1,X1,Z1)\n",
      "plt.xlabel(\"Length x in mm\")\n",
      "plt.ylabel(\"Radial Stress Distribution & Hoop Stress Distribution in N/mm**2\")\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Max Hoop Stress Developed in the cross-section is 28.0 N/mm**2\n",
        "Plot of Variation of hoop stress\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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29vaQJAknTpx4bAP379/HiBEjMGzYMMycOdNwT41GA1dXV2RkZGDgwIEcMiIi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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x563ec90>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.12,Page No.339"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d_o=300 #mm #Outside diameter \n",
      "d2=200 #mm #Internal Diameter\n",
      "p=12 #N/mm**2 #internal Fluid pressure\n",
      "F_max=16 #N/mm**2 #Tensile stress\n",
      "r_o=150 #mm #Outside Diameter\n",
      "r2=100 #mm #Internal Diameter\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Let p_o be the External Pressure applied.\n",
      "#From LLame's theorem\n",
      "#p_x=b*(x**2)**-1-a   ..............(1)\n",
      "#F_x=b*(x**2)**-1+a   ...........................(2)\n",
      "\n",
      "#Now At\n",
      "x=100 #mm\n",
      "p_x=12 #N/mm**2\n",
      "#sub in equation 1 we get\n",
      "#12=b*(100**2)**-1-a    . ..................(3)\n",
      "\n",
      "#The Max Hoop stress occurs at least value of x where\n",
      "x=r1=100 #mm\n",
      "#16=b*(100**2)**-1+a      .......................(4)\n",
      "\n",
      "#From Equations 1 and 2 we get\n",
      "#28=b*(100**2)**-1+b*(100**2)**-1\n",
      "#After furhter Simplifying we get\n",
      "b=28*100**2*2**-1\n",
      "\n",
      "#sub in equation 1 we get\n",
      "a=-(12-(b*(100**2)**-1))\n",
      "\n",
      "#Thus At\n",
      "x2=150 #mm\n",
      "p_o=b*(x2**2)**-1-a\n",
      "\n",
      "#Result\n",
      "print\"Minimum External applied is\",round(p_o,2),\"N/mm**2\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Minimum External applied is 4.22 N/mm**2\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.13,Page No.340"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d1=160 #mm #Internal Diameter \n",
      "r1=80 #mm #External Diameter\n",
      "p1=40 #N/mm**2 #Internal Diameter\n",
      "P_max=120 #N/mm**2 #Allowable stress\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#From Lame's Equation we have\n",
      "#p_x=b*(x**2)**-1-a       ..........................(1)\n",
      "#F_x=b*(x**2)**-1+a      ...........................(2)\n",
      "\n",
      "#At \n",
      "x=r1=80 #N/mm**2 \n",
      "#Sub in equation 1 we get\n",
      "#120=b*(80**2)**-1+a     ........................(3)\n",
      "\n",
      "#The hoop tension at inner edge is max stress\n",
      "#Hence\n",
      "#120=b*(80**2)**-1+a    .............................(4)\n",
      "\n",
      "#From Equation 3 and 4 we get\n",
      "b=160*80**2*2**-1 \n",
      "\n",
      "#Sub in equation 3 we get\n",
      "a=-(40-(b*(80**2)**-1))\n",
      "\n",
      "#Let External radius be r_o.Since at External Surface is Zero,we get\n",
      "#0=b*(r_o)**-1-a\n",
      "#After Further simplifying we get\n",
      "r_o=(b*a**-1)**0.5\n",
      "\n",
      "#Thickness of Cyclinder \n",
      "t=r_o-r1 #mm\n",
      "\n",
      "#Result\n",
      "print\"Thickness Required is\",round(t,2),\"mm\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Thickness Required is 33.14 mm\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.14,Page No.341"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d_o=300 #mm #Outside diameter \n",
      "d1=180 #mm #Internal Diameter\n",
      "p=12 #N/mm**2 #internal Fluid pressure\n",
      "p_o=6 #N/mm**2 #External Pressure\n",
      "r_o=150 #mm #Outside Diameter\n",
      "r=90 #mm #Internal Diameter\n",
      "\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#From Lame's Equation we have\n",
      "#p_x=b*(x**2)**-1-a       ..........................(1)\n",
      "#F_x=b*(x**2)**-1+a      ...........................(2)\n",
      "\n",
      "#At \n",
      "x=r1=90 #N/mm**2 \n",
      "p=42 #N/mm**2\n",
      "#Sub in equation 1 we get\n",
      "#42=b*(90**2)**-1-a       ..............................(3)\n",
      "\n",
      "#At \n",
      "x=r_o=150 #mm\n",
      "p2=6 #N/mm**2\n",
      "#sub in equation 1 we get\n",
      "#6=b*(150**2)**-1-a   ..............................(4)\n",
      "\n",
      "#From equations 3 and 4  weget\n",
      "#36=b*(90**2)**-1-b2(150**2)**-1\n",
      "#After further simplifying we get\n",
      "b=36*90**2*150**2*(150**2-90**2)**-1\n",
      "\n",
      "#Sub value of b in equation 4 we get\n",
      "a=b*(150**2)**-1-p_o\n",
      "\n",
      "#At \n",
      "x=r1=90 #mm\n",
      "F_x=b*(x**2)**-1+a #N/mm**2\n",
      "\n",
      "#At \n",
      "x2=r_o=150 #mm \n",
      "F_x2=b*(x2**2)**-1+a  #N/mm**2\n",
      "\n",
      "#Now if External pressure is doubled i.e p_o2=12 #N/mm**2 We have\n",
      "p_o2=12 #N/mm**2\n",
      "#sub in equation 4 we get\n",
      "#12=b2*(150**2)**-1-a2                    ..........................(5)\n",
      "\n",
      "#Max Hoop stress is to be 70.5 #N/mm**2,which occurs at x=r1=90 #mm\n",
      "#Sub in equation 4 we get\n",
      "#70.5=b*(90**2)**-1+a2           ................................(6)\n",
      "\n",
      "#Adding equation 5 and 6\n",
      "#82.5=b2*(150**2)**-1+b*(90**2)**-1\n",
      "#After furhter simplifying we get\n",
      "b2=82.5*150**2*90**2*(150**2+90**2)**-1\n",
      "\n",
      "#Sub in equation 5 we get\n",
      "a2=b2*(150**2)**-1-12 \n",
      "\n",
      "#If p_i is the internal pressure required then from Lame's theorem\n",
      "p_i=b2*(r1**2)**-1-a2\n",
      "\n",
      "#Result\n",
      "print\"Stresses int the material are:F_x\",round(F_x,2),\"N/mm**2\"\n",
      "print\"                             :F_x2\",round(F_x2,2),\"N/mm**2\"\n",
      "print\"Internal Pressure that can be maintained is\",round(p_i,2),\"N/mm**2\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Stresses int the material are:F_x 70.5 N/mm**2\n",
        "                             :F_x2 34.5 N/mm**2\n",
        "Internal Pressure that can be maintained is 50.82 N/mm**2\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.15,Page No.344"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "%matplotlib  inline\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "r1=200 #mm #Inner Radius\n",
      "r2=250 #mm #Radius at common surface\n",
      "r3=300 #mm #Outer radius\n",
      "p=6 #N/mm**2 #Inital pressure\n",
      "p2=80 #N/mm**2 #Pressure\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Inner Cyclinder:\n",
      "\n",
      "#From Lame's Equation we have\n",
      "#p_x=b*(x**2)**-1-a       ..........................(1)\n",
      "#F_x=b*(x**2)**-1+a      ...........................(2)\n",
      "\n",
      "#At \n",
      "x=r1=200 #mm\n",
      "p_x=0\n",
      "#0=b1*(250**2)**-1-a1   .................(3)\n",
      "\n",
      "#At x=r2=250 #mm\n",
      "p_x2=6 #N/mm**2\n",
      "#6=b1*(250**2)-a1    ...................(4)\n",
      "\n",
      "#From Equation 3 and 4 we get\n",
      "b1=6*200**2*250**2*(200**2-250**2)**-1\n",
      "\n",
      "#From equation 3 we get\n",
      "a1=b1*(200**2)**-1\n",
      "\n",
      "F_200=b1*(200**2)**-1+a1\n",
      "F_250=b1*(250**2)**-1+a1\n",
      "\n",
      "#For outer cyclinder \n",
      "#From Lame's Equation we have\n",
      "#p_x2=b2*(x**2)**-1-a2       ..........................(5)\n",
      "#F_x2=b2*(x**2)**-1+a2      ...........................(6)\n",
      "\n",
      "\n",
      "#At \n",
      "x2=r2=250 #mm\n",
      "p_x2=6 #N/mm**2\n",
      "#6=b2*(250**2)**-1-a2     ...........................(7)      \n",
      "\n",
      "#At\n",
      "x3=300 #mm\n",
      "#p_x2=0\n",
      "#0=b2**2*(300**2)**-1-a2     .................................(8)\n",
      "\n",
      "#from equation 7 and 8 we get\n",
      "b2=6*250**2*300**2*(300**2-250**2)**-1\n",
      "\n",
      "#sub in equation 8 we get\n",
      "a2=b2*(300**2)**-1\n",
      "\n",
      "F_250_2=b2*(250**2)**-1+a2\n",
      "F_300_2=b2*(300**2)**-1+a2\n",
      "\n",
      "#When Fluid is admitted\n",
      "#Let Lame's equation be\n",
      "#p_x3=b3*(x**2)**-1-a3       ..........................(5)\n",
      "#F_x3=b3*(x**2)**-1+a3      ...........................(6)\n",
      "\n",
      "\n",
      "#At x=200\n",
      "p_x3=80 #N/mm**2\n",
      "#80=b3*(200**2)**-1-a3    ................................(7)\n",
      "\n",
      "#At x=300 #mm\n",
      "#p_x=0\n",
      "#0=b3*(300**2)**-1-a3     ..............................(8)\n",
      "\n",
      "#from Equation 7 and 8 we get\n",
      "b3=80*200**2*300**2*(300**2-200**2)**-1\n",
      "\n",
      "#From Equation 8 we get\n",
      "a3=b3*(300**2)**-1\n",
      "\n",
      "#Hoop stresses \n",
      "F_200_3=b3*(200**2)**-1+a3 #N/mm**2\n",
      "F_250_3=b3*(250**2)**-1+a3 #N/mm**2\n",
      "F_300_3=b3*(300**2)**-1+a3 #N/mm**2\n",
      "\n",
      "#Pressure at common surface\n",
      "p_250=b3*(250**2)**-1-a3 #N/mm**2\n",
      "\n",
      "#final stress\n",
      "f_200=F_200+F_200_3 #N/mm**2\n",
      "f_250=F_250+F_250_3 #N/mm**2\n",
      "f_300=F_250_2+F_250_3 #N/mm**2\n",
      "f_300_2=F_300_2+F_300_3 #N/mm**2\n",
      "\n",
      "#Result\n",
      "print\"final Hoop stress are:f_200\",round(f_200,2),\"N/mm**2\"\n",
      "print\"                     :f_250\",round(f_250,2),\"N/mm**2\"\n",
      "print\"                     :f_300\",round(f_300,2),\"N/mm**2\"\n",
      "print\"                     :f_300_2\",round(f_300_2,2),\"N/mm**2\"\n",
      "print\"Variation of Hoop stress and Radial stress\"\n",
      "\n",
      "#Final stresses\n",
      "#Variation of hoop stress \n",
      " \n",
      "X1=[x,x2,x3,x3]\n",
      "Y1=[f_200,f_250,f_300,f_300_2]\n",
      "Z1=[0,0,0,0]\n",
      "plt.plot(X1,Y1,X1,Z1)\n",
      "plt.xlabel(\"Length x in mm\")\n",
      "plt.ylabel(\"Hoop Stress Distribution in N/mm**2\")\n",
      "plt.show()\n",
      "\n",
      "#Due to Fluid\n",
      "#Variation of hoop stress \n",
      " \n",
      "X1=[x,x2,x3]\n",
      "Y1=[F_200_3,F_250_3,F_300_3]\n",
      "Z1=[0,0,0]\n",
      "plt.plot(X1,Y1,X1,Z1)\n",
      "plt.xlabel(\"Length x in mm\")\n",
      "plt.ylabel(\"Hoop Stress Distribution in N/mm**2\")\n",
      "plt.show()\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "final Hoop stress are:f_200 174.67 N/mm**2\n",
        "                     :f_250 128.83 N/mm**2\n",
        "                     :f_300 189.43 N/mm**2\n",
        "                     :f_300_2 155.27 N/mm**2\n",
        "Variation of Hoop stress and Radial stress\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MHz4cr7zyCmbPno1///vf1fIBnZeXp/l+8+bNmhVR0dHR+P7771FYWIjMzExcvHgRnTp1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\nEenEREFERDoxURARkU7/H21GIMqBrUIbAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5576590>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x566a330>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.16,Page No.348"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "do=200 #mm #Inner Diameter\n",
      "r_o=100 #mm #Inner radius\n",
      "d1=300 #mm #outer diameter\n",
      "r1=150 #mm #Outer radius\n",
      "d2=250 #mm #Junction Diameter\n",
      "r2=125 #mm #Junction radius\n",
      "E=2*10**5 #N/mm**2 #Modulus of Elasticity\n",
      "p=30 #N/mm**2 #radial pressure\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#from Lame's Equation we get\n",
      "#p_x=b*(x**2)**-1-a       ..........................(1)\n",
      "#F_x=b*(x**2)**-1+a      ...........................(2)\n",
      "\n",
      "#Then from Boundary condition \n",
      "#p_x=0 at x=100 #mm\n",
      "#0=b1*(100**2)**-1-a1     .....................(3)\n",
      "\n",
      "#p_x2=30 #N/mm**2 at x2=125 #mm\n",
      "#30=b1*(125**2)**-1-a1    ................................(4)\n",
      "\n",
      "#From equation 3 and 4 we get\n",
      "b1=30*125**2*100**2*(100**2-125**2)**-1\n",
      "\n",
      "#From Equation 3 we get\n",
      "a1=b1*(100**2)**-1\n",
      "\n",
      "#therefore Hoop stress in inner cyclinder at junction\n",
      "F_2_1=b1*(125**2)**-1+a1 #N/mm**2\n",
      "\n",
      "#Outer Cyclinder\n",
      "#p_x=b*(x**2)**-1-a       ..........................(5)\n",
      "#F_x=b*(x**2)**-1+a      ...........................(6)\n",
      "\n",
      "#Now at x=125 #mm\n",
      "#p_x3=30 #N/mm**2\n",
      "#30=b2*(125**2)**-1-a2     ..................................(7)\n",
      "\n",
      "#At x=150 #mm\n",
      "#p_x4=0\n",
      "#0=b2*(150**2)**-1-a2     ...................................(8)\n",
      "\n",
      "#From equations 7 and 8\n",
      "b2=30*150**2*125**2*(150**2-125**2)**-1\n",
      "\n",
      "#From eqauation 8 we get\n",
      "a2=b2*(150**2)**-1\n",
      "\n",
      "#Hoop stress at junction \n",
      "F_2_0=b2*(125**2)**-1+a2 #N/mm**2\n",
      "\n",
      "rho_r=(F_2_0-F_2_1)*E**-1*r2\n",
      "\n",
      "#Result\n",
      "print\"Shrinkage Allowance is\",round(rho_r,3),\"mm\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Shrinkage Allowance is 0.189 mm\n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.17,Page No.350"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d_o=500 #mm #Outer Diameter\n",
      "r_o=250 #mm #Outer Radius\n",
      "d1=300 #mm #Inner Diameter\n",
      "r1=150 #mm #Inner Radius\n",
      "d2=400 #mm #Junction Diameter\n",
      "E=2*10**5 #N/mm**2 #Modulus ofElasticity\n",
      "alpha=12*10**-6 #Per degree celsius\n",
      "dell_d=0.2 #mm\n",
      "dell_r=0.1 #mm\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#Let p be the radial pressure developed at junction\n",
      "#Let Lame's Equation for internal cyclinder be\n",
      "#p_x=b*(x**2)**-1-a   ................................(1)\n",
      "#F_x=b*(x**2)**-1+a     ...............................(2)\n",
      "\n",
      "#At \n",
      "x=150 #mm \n",
      "p_x=0\n",
      "#Sub in equation 1 we get\n",
      "#0=b*(150**2)**-1-a    .........................(3)\n",
      "\n",
      "#At \n",
      "x2=200 #mm\n",
      "#p_x2=p\n",
      "#p=b*(200**2)**-1-a    ......................(4)\n",
      "  \n",
      "#From Equation 3 and 4\n",
      "#p=b*(200**2)**-1-b(150**2)**-1\n",
      "#after further simplifying we get\n",
      "#b=-51428.571*p\n",
      "\n",
      "#sub in equation 3 we get\n",
      "#a1=-2.2857*p\n",
      "\n",
      "#therefore hoop stress at junction is\n",
      "#F_2_1=-21428.571*p*(200**2)**-1-2.2857*p\n",
      "#after Further simplifying we geet\n",
      "#F_2_1=3.5714*p\n",
      "\n",
      "#Let Lame's Equation for cyclinder be \n",
      "#p_x=b*(x**2)**-1-a    .........................5\n",
      "#F_x=b*(x**2)**-1+a   .............................6\n",
      "\n",
      "#At \n",
      "x=200 #mm\n",
      "#p_x=p2\n",
      "#p2=b2*(20**2)**-1-a2    ...................7\n",
      "\n",
      "#At\n",
      "x2=200 #mm\n",
      "p_x2=0\n",
      "#0=b2*(250**2)**-1-a2    ....................8\n",
      "\n",
      "#from equation 7 and 8 we get\n",
      "#p2=b2*(200**2)**-1-b2*(250**2)**-1\n",
      "#After further simplifying we get\n",
      "#p2=b2*(250**2-200**2)*(200**2*250**2)**-1\n",
      "#b2=111111.11*p\n",
      "\n",
      "#from equation 7\n",
      "#a2=b2*(250**2)**-1\n",
      "#further simplifying we get\n",
      "#a2=1.778*p\n",
      "\n",
      "#At the junctionhoop stress in outer cyclinder \n",
      "#F_2_0=b2*(200**2)**-1+a2\n",
      "#After further simplifying we get\n",
      "#F_2_0=4.5556*p\n",
      "\n",
      "#Considering circumferential strain,the compatibility condition\n",
      "#rho_r*r2**-1=1*E**-1*(F_2_1+F_2_0)\n",
      "#where F_2_1 is compressive and F_2_0 is tensile\n",
      "#furter simplifying we get\n",
      "p=0.1*200**-1*2*10**5*(3.5714+4.5556)**-1\n",
      "\n",
      "#Let T be the rise in temperature required\n",
      "#dell_d=d*alpha*T\n",
      "#After sub values and further simplifying we get\n",
      "d=250 #mm\n",
      "T=dell_d*(d*alpha)**-1 #Per degree celsius\n",
      "\n",
      "#Result\n",
      "print\"Radial Pressure Developed at junction\",round(p,2),\"N/mm**2\"\n",
      "print\"Min Temperatureto outer cyclinder\",round(T,2),\"Per degree Celsius\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Radial Pressure Developed at junction 12.3 N/mm**2\n",
        "Min Temperatureto outer cyclinder 66.67 Per degree Celsius\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Example 8.8.18,Page No.355"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import numpy as np\n",
      "%matplotlib  inline\n",
      "\n",
      "#Initilization of Variables\n",
      "\n",
      "d_o=400 #mm #Outer Diameter\n",
      "r_o=200 #mm #Outer radius\n",
      "t=50 #mm #Thickness\n",
      "r1=150 #mm #Internal Radius\n",
      "p=50 #N/mm**2 #Internal Pressure\n",
      "\n",
      "#Calculations\n",
      "\n",
      "#The Radial Pressure and hoop stress at any radial distance x are given by\n",
      "#p_x=b*(x**2)**-1-a       ..........................(1)\n",
      "#F_x=b*(x**2)**-1+a      ...........................(2)\n",
      "\n",
      "#Now at\n",
      "x=150 #N/mm**2\n",
      "p_x1=50 #N/mm**2\n",
      "#Sub in equation 1 we get\n",
      "#50=2*b*(150**3)**-1-a    ...........................(3)\n",
      "\n",
      "#At x=200 #mm\n",
      "p_x2=0\n",
      "#0=2*b*(200**2)**-1-a     ....................(4)\n",
      "\n",
      "#From equation 3 and 4 we get\n",
      "#50=2*b*(150**3)**-1-2*b*(200**3)**-1\n",
      "#After further simplifying we get\n",
      "b=50*150**3*200**3*(200**3-150**3)**-1*2**-1\n",
      "\n",
      "#Sub in equation 3 we get\n",
      "a=b*(200**3)**-1\n",
      "\n",
      "#Now At\n",
      "x=150 #mm\n",
      "F_x=b*(x**3)**-1+a\n",
      "\n",
      "#Now At\n",
      "x2=160 #mm\n",
      "F_x2=b*(x2**3)**-1+a\n",
      "\n",
      "#Now At\n",
      "x3=170 #mm\n",
      "F_x3=b*(x3**3)**-1+a\n",
      "\n",
      "#Now At\n",
      "x4=180 #mm\n",
      "F_x4=b*(x4**3)**-1+a\n",
      "\n",
      "#Now At\n",
      "x5=190 #mm\n",
      "F_x5=b*(x5**3)**-1+a\n",
      "\n",
      "#Now At\n",
      "x6=200 #mm\n",
      "F_x6=b*(x6**3)**-1+a\n",
      "\n",
      "#Result\n",
      "print\"Plot of Variation of hoop stress\"\n",
      "\n",
      "#Plotting Variation of hoop stress\n",
      "\n",
      "X1=[x,x2,x3,x4,x5,x6]\n",
      "Y1=[F_x,F_x2,F_x3,F_x4,F_x5,F_x6]\n",
      "Z1=[0,0,0,0,0,0]\n",
      "plt.plot(X1,Y1,X1,Z1)\n",
      "plt.xlabel(\"Length x in mm\")\n",
      "plt.ylabel(\"Hoop Stress Distribution in N/mm**2\")\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Plot of Variation of hoop stress\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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yhIaGAgB8fX01nyMyFgOBnhrl5eVo164dTp8+Xef71ffEFxVTYwqF\nosaT+EQ9U2aVT6gDgBYtWmi9BJOSkoKOHTvq/eCVuup6UsuWLWu0XfmdJ+uoflzfeon0xTkEemq0\nbdsWPXv2xP/+7/8CkH65pqSk1PudoUOHYuvWrRBCICcnR7N9NCBNIj948MCgGjIyMvD111/j9OnT\n2LNnT52PI6wvdBpCrvMSVWIgkMUqLCxEt27dNK9vv/0WGzZswKpVq+Dv7w8fHx/s2rVL8/nqT4Kq\n/HnChAlwdnZGnz59MHXqVPTr1w/29vYAgNmzZ+Oll17STCo/+f0nnywlhMDMmTOxdOlSODk5YdWq\nVZg5c6bmspW272r7+cnvaPtz5c/1nbe+cxPpi8tOqckrKChA69atce/ePQwePBj//ve/azyhj4gk\nnEOgJm/MmDHIz89HcXExPv74Y4YBkRYcIRAREQDOIRARUQUGAhERAWAgEBFRBQYCEREBYCAQEVEF\nBgIREQEA/h+bezx5xlsz+QAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x58d75b0>"
       ]
      }
     ],
     "prompt_number": 18
    }
   ],
   "metadata": {}
  }
 ]
}