path: root/Engineering_Mechanics_by_Tayal_A.K./chapter12_11.ipynb

{
 "metadata": {
  "name": "chapter12.ipynb"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 12: Moment Of Inertia"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 12.12-7,Page No:285"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "# Initilization of variables\n",
      "\n",
      "A= 50 # cm^2 # area of the shaded portion\n",
      "J_A=22.5*10**2 # cm^4 # polar moment of inertia of the shaded portion\n",
      "d=6 # cm\n",
      "\n",
      "# Calculations\n",
      "\n",
      "J_c=J_A-(A*d**2) \n",
      "\n",
      "# substuting the value of I_x from eq'n 2 in eq'n 1 we get,\n",
      "I_y=J_c*0.333 # cm^4 # M.O.I about Y-axis # 1/3=0.333\n",
      "\n",
      "# Now from eq'n 2,\n",
      "I_x=2*I_y # cm^4 # M.O.I about X-axis\n",
      "\n",
      "# Results\n",
      "\n",
      "print\"The centroidal moment of inertia about X-axis (I_x) is \",round(I_x),\"cm^4\"\n",
      "print\"The centroidal moment of inertia about Y-axis (I_y) is \",round(I_y),\"cm^4\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The centroidal moment of inertia about X-axis (I_x) is  300.0 cm^4\n",
        "The centroidal moment of inertia about Y-axis (I_y) is  150.0 cm^4\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 12.12-8,Page No:288"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "# Initilization of variables\n",
      "\n",
      "b=20 # cm # width of the pate\n",
      "d=30 # cm # depth of the plate\n",
      "r=15 # cm # radius of the circular hole\n",
      "h=20 # cm # distance between the centre of the circle & the x-axis\n",
      "\n",
      "# Calculations\n",
      "\n",
      "# (a) Location of the centroid of the composite area\n",
      "\n",
      "A_1=b*d # cm^2 # area of the plate\n",
      "y_1=d/2 # cm # y-coordinate of the centroid\n",
      "A_2=(pi*r**2)/4 # cm^2 # area of the circle removed (negative)\n",
      "y_2=h # cm # y-coordinate of the centroid\n",
      "y_c=((A_1*y_1)-(A_2*y_2))/(A_1-A_2) # cm # from the bottom edge\n",
      "\n",
      "# (b) Moment of Inertia of the composite area about the centroidal x-axis\n",
      "\n",
      "# Area (A_1) M.I of area A_1 about x-axis\n",
      "I_x1=(b*(d**3))/12 # cm^4\n",
      "\n",
      "# M.I of the area A_1 about the centroidal x-axis of the composite area (By parallel-axis theorem)\n",
      "OC_1=15 # cm # from the bottom edge\n",
      "OC_2=20 # cm\n",
      "OC=12.9 # cm # from the bottom edge\n",
      "d_1=OC_1-OC # cm\n",
      "d_2=OC_2-OC # cm \n",
      "I_X1=(I_x1)+(A_1*d_1**2) # cm^4\n",
      "\n",
      "# Area(A_2) M.I of area A_2 about x-axis\n",
      "I_x2=(pi*r**4)/64 # cm^2\n",
      "\n",
      "#M.I of the area A_2 about the centroidal x-axis of the composite area (By parallel-axis theorem)\n",
      "I_X2=(I_x2)+(A_2*d_2**2) # cm^4\n",
      "\n",
      "# COMPOSITE AREA:M.O.I of the composite area about the centroidal x-axis\n",
      "I_x=(I_X1)-(I_X2) # cm^4\n",
      "\n",
      "# Results\n",
      "\n",
      "print\"The M.O.I of the composite area about the centroidal x-axis is \",round(I_x),\"cm^4\" \n",
      "#due to decimal variance answer varies by 1cm^4 from textbook"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The M.O.I of the composite area about the centroidal x-axis is  36253.0 cm^4\n"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 12.12-9,Page No:289"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "# Initilization of variables\n",
      "\n",
      "b1=80 # mm # width of the flange pate\n",
      "d1=20 # mm # depth of the flange plate\n",
      "b2=40 # mm # width/thickness of the web\n",
      "d2=60 # mm # depth of the web\n",
      "\n",
      "# Calculations\n",
      "\n",
      "# (a) Location of the centroid of the composite area\n",
      "A_1=b1*d1 # mm^2 # area of the flange plate\n",
      "y_1=d2+(d1/2) # mm # y-coordinate of the centroid\n",
      "A_2=b2*d2 # mm^2 # area of the web\n",
      "y_2=d2/2 # mm # y-coordinate of the centroid\n",
      "y_c=((A_1*y_1)+(A_2*y_2))/(A_1+A_2) # mm # from the bottom edge\n",
      "\n",
      "# (b) Moment of Inertia of the composite area about the centroidal x-axis\n",
      "\n",
      "# Area (A_1) M.I of area A_1 about x-axis\n",
      "I_x1=(b1*(d1**3))/12 # mm^4\n",
      "# M.I of the area A_1 about the centroidal x-axis of the composite area (By parallel-axis theorem)\n",
      "OC_1=70 # mm # from the bottom edge\n",
      "OC_2=30 # mm # from the bottom edge\n",
      "OC=y_c # mm # from the bottom edge\n",
      "d_1=(d2-y_c)+(d1/2) # mm\n",
      "d_2=y_c-OC_2 # mm \n",
      "I_X1=(I_x1)+(A_1*d_1**2) # mm^4\n",
      "\n",
      "# Area(A_2) M.I of area A_2 about x-axis\n",
      "I_x2=(b2*d2**3)/12 # mm^4\n",
      "# M.I of the area A_2 about the centroidal x-axis of the composite area (By parallel-axis theorem)\n",
      "I_X2=(I_x2)+(A_2*d_2**2) # mm^4\n",
      "# COMPOSITE AREA:M.O.I of the composite area about the centroidal x-axis\n",
      "I_x=(I_X1)+(I_X2) # mm^4\n",
      "\n",
      "# Results\n",
      "\n",
      "print\"The M.O.I of the composite area about the centroidal x-axis is \",round(I_x),\"mm^4\"\n",
      "# NOTE: The answer given in the text book is 2.31*10^3 insted of 2.31*10^6.\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The M.O.I of the composite area about the centroidal x-axis is  2309333.0 mm^4\n"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 12.12-10,Page No:291"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "import math\n",
      "\n",
      "# Initilization of variables\n",
      "b1=120 # mm # width of the flange pate of L-section\n",
      "d1=20 # mm # depth of the flange plate\n",
      "b2=20 # mm # width/thickness of the web\n",
      "d2=130 # mm # depth of the web\n",
      "\n",
      "# Calculations\n",
      "\n",
      "# (a) Location of the centroid of the composite area\n",
      "A_1=b1*d1 # mm^2 # area of the flange plate\n",
      "A_2=b2*d2 # mm^2 # area of the web\n",
      "y_1=d2+(d1/2) # mm # y-coordinate of the centroid\n",
      "y_2=d2/2 # mm # y-coordinate of the centroid\n",
      "x_1=60 # mm # x-coordinate of the centroid\n",
      "x_2=110 # mm # x-coordinate of the centroid\n",
      "y_c=((A_1*y_1)+(A_2*y_2))/(A_1+A_2) # mm # from the bottom edge\n",
      "x_c=((A_1*x_1)+(A_2*x_2))/(A_1+A_2) # mm # from the bottom edge\n",
      "\n",
      "# (b) Moment of Inertia of the composite area about the centroidal x-axis\n",
      "\n",
      "# Area (A_1) M.I of area A_1 about x-axis\n",
      "I_x1=(b1*(d1**3))/12 # mm^4\n",
      "# M.I of the area A_1 about the centroidal x-axis of the composite area (By parallel-axis theorem)\n",
      "OC_1=d2+(d1/2) # mm # from the bottom edge\n",
      "OC_2=d2/2 # mm # from the bottom edge\n",
      "OC=y_c # mm # from the bottom edge\n",
      "d_1=(d2-y_c)+(d1/2) # mm\n",
      "d_2=y_c-OC_2 # mm \n",
      "I_X1=(I_x1)+(A_1*d_1**2) # mm^4\n",
      "\n",
      "# Area(A_2) M.I of area A_2 about x-axis\n",
      "I_x2=(b2*d2**3)/12 # mm^4\n",
      "\n",
      "# M.I of the area A_2 about the centroidal x-axis of the composite area (By parallel-axis theorem)\n",
      "I_X2=(I_x2)+(A_2*d_2**2) # mm^4\n",
      "# COMPOSITE AREA:M.O.I of the composite area about the centroidal x-axis\n",
      "I_x=(I_X1)+(I_X2) # mm^4\n",
      "\n",
      "# (c) Moment of Inertia of the composite area about the centroidal y-axis\n",
      "\n",
      "# Area (A_1) M.I of area A_1 about y-axis\n",
      "I_y1=(d1*(b1**3))/12 # mm^4\n",
      "# M.I of the area A_1 about the centroidal y-axis of the composite area (By parallel-axis theorem)\n",
      "d_3=x_c-(b1/2) # mm # distance between c &c1 along x axis\n",
      "I_Y1=(I_y1)+(A_1*d_3**2) # mm^4\n",
      "\n",
      "# Area(A_2) M.I of area A_2 about y-axis\n",
      "I_y2=(d2*b2**3)/12 # mm^4\n",
      "\n",
      "# M.I of the area A_2 about the centroidal y-axis of the composite area (By parallel-axis theorem)\n",
      "d_4=b1-x_c-(b2/2) # mm # distance between c &c2 along x axis\n",
      "I_Y2=(I_y2)+(A_2*d_4**2) # mm^4\n",
      "# COMPOSITE AREA:M.O.I of the composite area about the centroidal y-axis\n",
      "I_y=(I_Y1)+(I_Y2) # mm^4\n",
      "\n",
      "# Results\n",
      "\n",
      "print\"The M.O.I of the composite area about the centroidal x-axis is \",round(I_x),\"mm^4\"\n",
      "print\"The M.O.I of the composite area about the centroidal Y-axis is \",round(I_y),\"mm^4\"\n",
      "# NOTE: The answer for I_x given in text book is 0.76*10^6 insted of 10.76*10^6\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The M.O.I of the composite area about the centroidal x-axis is  10761666.0 mm^4\n",
        "The M.O.I of the composite area about the centroidal Y-axis is  6086666.0 mm^4\n"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 12.12-14,Page No:299"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "# Initilization of variables\n",
      "\n",
      "b=1 # cm # smaller side of the L-section\n",
      "h=4 # cm # larger side of the L-section\n",
      "\n",
      "# Calculations\n",
      "\n",
      "# (A) RECTANGLE A_1: Using the paralel axis theorem\n",
      "Ixy=0\n",
      "I_xy1=(Ixy)+((h*b)*(b*0.5)*(h*0.5)) # cm**4\n",
      "\n",
      "# (B) RECTANGLE A_2: Using the paralel axis theorem\n",
      "I_xy2=(Ixy)+((b*(h-1))*(1+(1.5))*(b*0.5)) # cm**4\n",
      "\n",
      "# Product of inertia of the total area\n",
      "I_xy=I_xy1+I_xy2 # cm**4\n",
      "\n",
      "# Results\n",
      "\n",
      "print\"The Product of inertia of the L-section is \",round(I_xy,2),\"cm^4\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The Product of inertia of the L-section is  7.75 cm^4\n"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 12.12-15,Page No:300"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "# Initilization of variables\n",
      "\n",
      "I_x=1548 # cm^4 # M.O.I of the Z-section about X-axis\n",
      "I_y=2668 # cm^4 # M.O.I of the Z-section about Y-axis\n",
      "b=12 # cm # width of flange of the Z-section\n",
      "d=3 # cm # depth of flange of the Z-section\n",
      "t=2 # cm # thickness of the web of the Z-section\n",
      "h=6 # cm # depth of the web of the Z-section\n",
      "\n",
      "#Calculations\n",
      "\n",
      "A_1=b*d # cm^2 # area of top flange\n",
      "x_1=-5 # cm # distance of the centroid from X-axis for top flange\n",
      "y_1=4.5 # cm # distance of the centroid from Y-axis for top flange\n",
      "A_2=t*h # cm^2 # area of web\n",
      "x_2=0 # cm # distance of the centroid from X-axis for the web\n",
      "y_2=0 # cm # distance of the centroid from Y-axis for the web\n",
      "A_3=b*d # cm^2 # area of bottom flange\n",
      "x_3=5 # cm # distance of the centroid from X-axis for top flange\n",
      "y_3=-4.5 # cm # distance of the centroid from Y-axis for top flange\n",
      "\n",
      "# Product of Inertia of the total area is,\n",
      "I_xy=((A_1*x_1*y_1)+(A_3*x_3*y_3)) # cm^4\n",
      "# The direction of the principal axes is,\n",
      "theta_m=(arctan((2*I_xy)/(I_y-I_x))*(180/pi))/2 # degree\n",
      "# Principa M.O.I\n",
      "I_max=((I_x+I_y)/2)+((((I_x-I_y)/2)**2+(I_xy)**2)**0.5) # cm**4\n",
      "I_mini=((I_x+I_y)/2)-((((I_x-I_y)/2)**2+(I_xy)**2)**0.5) # cm**4\n",
      "\n",
      "# Results\n",
      "\n",
      "print\"The principal axes of the section about O is \",round(theta_m,2),\"degree\"\n",
      "print\"The Maximum value of principal M.O.I is \",round(I_max),\"cm^4\"\n",
      "print\"The Minimum value of principal M.O.I is \",round(I_mini),\"cm^4\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The principal axes of the section about O is  -35.47 degree\n",
        "The Maximum value of principal M.O.I is  3822.0 cm^4\n",
        "The Minimum value of principal M.O.I is  394.0 cm^4\n"
       ]
      }
     ],
     "prompt_number": 50
    }
   ],
   "metadata": {}
  }
 ]
}