path: root/Mechanics_of_Materials/chapter7.ipynb

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 7: Analysis of Stress and Strain"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 7.1, page no. 472"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"\"\"\n",
      "calculate stresses acting on an element inclined at 45\u00b0\n",
      "\"\"\"\n",
      "\n",
      "import math \n",
      "\n",
      "#initialisation\n",
      "# Let x1, y1 be the transformed direction inclined at 45 deegree to the original\n",
      "sx = 16000                          # Direct stress in x-direction in psi\n",
      "sy = 6000                           # Direct stress in y-direction \"\"\n",
      "txy = 4000                          # Shear stress in y-direction \"\"\n",
      "tyx = txy                           # Shear stress in x-direction \"\"\n",
      "t = 45                              # Inclination pf plane in degree \n",
      "\n",
      "#calculation\n",
      "sx1 = (sx+sy)/2 + ((sx-sy)*(math.cos(math.radians(2*t))/2.0)) + txy*math.sin(math.radians(2*t))       # Direct stress in x1-direction in psi\n",
      "sy1 = (sx+sy)/2 - ((sx-sy)*(math.cos(math.radians(2*t))/2.0)) - txy*math.sin(math.radians(2*t))       # Direct stress in y1-direction in psi\n",
      "tx1y1 =  - ((sx-sy)*(math.sin(math.radians(2*t))/2.0)) + txy*math.cos(math.radians(2*t))    # Shear stress in psi\n",
      "\n",
      "print \"The direct stress on the element in x1-direction is\", sx1, \"psi\"\n",
      "print \"The direct stress on the element in y1-direction is\", sy1, \"psi\"\n",
      "print \"The shear stress on the element\", tx1y1, \"psi\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The direct stress on the element in x1-direction is 15000.0 psi\n",
        "The direct stress on the element in y1-direction is 7000.0 psi\n",
        "The shear stress on the element -5000.0 psi\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 7.2, page no. 473"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"\"\"\n",
      "stresses acting on an element that is oriented at a clockwise 15\u00b0 \n",
      "\"\"\"\n",
      "\n",
      "import math \n",
      "\n",
      "#initialisation\n",
      "# Let x1, y1 be the transformed direction inclined at 15 deegree to the original\n",
      "sx = -46e06                     # Direct stress in x-direction in Pa\n",
      "sy = 12e06                      # Direct stress in y-direction \"\"\n",
      "txy = -19e06                    # Shear stress in y-direction \"\"\n",
      "t = -15                         # Inclination of plane in degree \n",
      "\n",
      "#calculation\n",
      "sx1 = (sx+sy)/2.0 + ((sx-sy)*(math.cos(math.radians(2*t))/2.0)) + txy*math.sin(math.radians(2*t))   # Direct stress in x1-direction in Pa\n",
      "sy1 = (sx+sy)/2.0 - ((sx-sy)*(math.cos(math.radians(2*t))/2.0)) - txy*math.sin(math.radians(2*t))   # Direct stress in y1-direction in Pa\n",
      "tx1y1 = -((sx-sy)*(math.sin(math.radians(2*t))/2.0)) + txy*math.cos(math.radians(2*t))             # Shear stress in Pa\n",
      "\n",
      "\n",
      "print \"The direct stress on the element in x1-direction is\", sx1, \"Pa\"\n",
      "print \"The direct stress on the element in y1-direction is\", sy1, \"Pa\"\n",
      "print \"The shear stress on the element\", tx1y1, \"Pa\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The direct stress on the element in x1-direction is -32614736.7097 Pa\n",
        "The direct stress on the element in y1-direction is -1385263.29025 Pa\n",
        "The shear stress on the element -30954482.6719 Pa\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "example 7.3, page no. 481"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"\"\"\n",
      "Calculate shear stress and principal stress\n",
      "\"\"\"\n",
      "\n",
      "import math\n",
      "\n",
      "ax = 12300.0\n",
      "ay = -4200.0\n",
      "txy = -4700.0\n",
      "\n",
      "tan_2p = round((2*txy)/(ax-ay), 4)\n",
      "\n",
      "theta_p1 = 150.3\n",
      "theta_p2 = 330.3\n",
      "\n",
      "stress1 = (ax+ay)/2.0\n",
      "stress2 = (ax-ay)/2.0\n",
      "a1 = stress1 + math.sqrt((stress2**2.0)+(txy**2.0))\n",
      "a2 = stress1 - math.sqrt((stress2**2.0)+(txy**2.0))\n",
      "\n",
      "#python calculations differ a bit. hence, differences in the answer\n",
      "print \"Principal stesses are \", round(a1), \"psi and \", round(a2), \" psi\"\n",
      "\n",
      "tmax = math.sqrt((stress2**2.0)+(txy**2.0))\n",
      "print \"Maximum shear stress is \", round(tmax), \" psi\"\n",
      "\n",
      "a_aver = (ax+ay)/2.0\n",
      "\n",
      "print \"Normal stress acting at maximum shear stress = \", round(a_aver), \"psi\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Principal stesses are  13545.0 psi and  -5445.0  psi\n",
        "Maximum shear stress is  9495.0  psi\n",
        "Normal stress acting at maximum shear stress =  4050.0 psi\n"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 7.4, page no. 492"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"\"\"\n",
      "stresses acting on an element inclined at 30\u00b0 \n",
      "\"\"\"\n",
      "\n",
      "import math \n",
      "\n",
      "#initialisation\n",
      "sx = 90e06                      # Direct stress in x-direction in Pa\n",
      "sy = 20e06                      # Direct stress in y-direction in Pa\n",
      "t = 30                          # Inclination of element in degree\n",
      "\n",
      "#calculation\n",
      "savg = (sx+sy)/2.0                                  # Average in-plane direct stress\n",
      "txy = 0 \n",
      "R = math.sqrt(((sx-sy)/2)**2+(txy)**2)              # Radius of mohr circle\n",
      "\n",
      "# Point D  at 2t = 60\n",
      "sx1 = savg + R*math.cos(math.radians(2*t))          # Direct stress at point D \n",
      "tx1y1 = -R*math.sin(math.radians(2*t))              # shear stress at point D\n",
      "print \"The direct stress at point D is\", sx1, \"Pa\"\n",
      "print \"The shear stress at point D is\", tx1y1, \"Pa\"\n",
      "\n",
      "# Point D  at 2t = 240\n",
      "sx2 = savg + R*math.cos(math.radians(90 + t))       # Direct stress at point D \n",
      "tx2y2 = R*math.sin(math.radians(90 + t))            # shear stress at point D\n",
      "print \"The direct stress at point D_desh is\", sx2, \"Pa\"\n",
      "print \"The shear stress at point D_desh is\", tx2y2, \"Pa\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The direct stress at point D is 72500000.0 Pa\n",
        "The shear stress at point D is -30310889.1325 Pa\n",
        "The direct stress at point D_desh is 37500000.0 Pa\n",
        "The shear stress at point D_desh is 30310889.1325 Pa\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 7.5, page no. 494"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"\"\"\n",
      "stresses acting on an element, principal stress, max. shear stress\n",
      "\"\"\"\n",
      "\n",
      "import math\n",
      "import numpy\n",
      "\n",
      "#initialisation \n",
      "sx = 15000                              # Direct stress in x-direction in psi\n",
      "sy = 5000                               # Direct stress in y-direction \"\"\n",
      "txy = 4000                              # Shear stress in y-direction \"\"\n",
      "savg = (sx+sy)/2.0                        # Average in-plane direct stress\n",
      "sx1 = 15000                             # Stress acting on face at theta = 0 degree\n",
      "tx1y1 = 4000                            # Stress acting on face at theta = 0 degree\n",
      "sx1_ = 5000                             \n",
      "tx1y1_ = -4000                          \n",
      "\n",
      "#calculation\n",
      "R = math.sqrt(((sx-sy)/2)**2+(txy)**2)              # Radius of mohr circle\n",
      "\n",
      "# Part (a)\n",
      "t = 40                                              # Inclination of the plane in degree\n",
      "f1 = numpy.degrees(numpy.arctan((4000.0/5000.0)))     # Angle between line CD and x1-axis\n",
      "f2 = 80 - f1                                        # Angle between line CA and x1-axis\n",
      "\n",
      "# Point D  \n",
      "sx1 = savg + R*math.cos(math.radians(f2))       # Direct stress at point D \n",
      "tx1y1 = -R*math.sin(math.radians(f2))           # shear stress at point D\n",
      "print \"The shear stress at point D\", round(tx1y1), \"psi\"\n",
      "\n",
      "# Point D'  \n",
      "sx2 = savg - R*math.cos(math.radians(f2))       # Direct stress at point D' \n",
      "tx2y2 = R*math.sin(math.radians(f2))            # shear stress at point D'\n",
      "print \"The direct stres at point D_desh\", round(sx2), \"psi\"\n",
      "\n",
      "#Part (b)\n",
      "sp1 = savg + R                                      # Maximum direct stress in mohe circle (at point P1)\n",
      "tp1 = f1/2                                          # Inclination of plane of maximum direct stress\n",
      "print \"with angle\",round(tp1,2) , \"degree The maximum direct stress at P1 is \",sp1 , \"psi\"\n",
      "sp2 =  savg - R  # Minimum direct stress in mohe circle (at point P2)\n",
      "tp2 = (f1+180)/2  # Inclination of plane of minimum direct stress\n",
      "print \"with angle\", round(tp2,2), \"degree The maximum direct stress at P2 is \",sp2 , \"psi\"\n",
      "\n",
      "# Part (c)\n",
      "tmax = R                                            # Maximum shear stress in mohe circle\n",
      "ts1 = -(90 - f1)/2.0                                # Inclination of plane of maximum shear stress\n",
      "print \"with plane incilation of\", tmax, \"psi The Maximum shear stress is \", round(ts1,2), \"deegree\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The shear stress at point D -4229.0 psi\n",
        "The direct stres at point D_desh 5193.0 psi\n",
        "with angle 19.33 degree The maximum direct stress at P1 is  16403.1242374 psi\n",
        "with angle 109.33 degree The maximum direct stress at P2 is  3596.87576257 psi\n",
        "with plane incilation of 6403.12423743 psi The Maximum shear stress is  -25.67 deegree\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 7.6, Page number 497"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"\"\"\n",
      "use Mohr\u2019s circle, to calculate various quantities\n",
      "\"\"\"\n",
      "\n",
      "import math \n",
      "import numpy\n",
      "\n",
      "\n",
      "sx = -50e06                                 # Direct stress in x-direction in psi\n",
      "sy = 10e06                                  # Direct stress in y-direction \"\"\n",
      "txy = -40e06                                # Shear stress in y-direction \"\"\n",
      "savg = (sx+sy)/2                            # Average in-plane direct stress\n",
      "sx1 = -50e06\n",
      "tx1y1 = -40e06                              # Stress acting on face at theta = 0 degree\n",
      "sx1_ = 10e06\n",
      "tx1y1_ = 40e06                              # Stress acting on face at theta = 0 degree\n",
      "\n",
      "#calculation\n",
      "R = math.sqrt(((sx-sy)/2)**2+(txy)**2)      # Radius of mohr circle\n",
      "\n",
      "# Part (a)\n",
      "t = 45                                                  # Inclination of the plane in degree\n",
      "f1 = numpy.degrees(numpy.arctan((40e06/30e06)))       # Angle between line CD and x1-axis\n",
      "f2 = 90 - f1  # Angle between line CA and x1-axis\n",
      "\n",
      "# Point D  \n",
      "sx1 = savg - R*math.cos(math.radians(f2))           # Direct stress at point D \n",
      "tx1y1 = R*math.sin(math.radians(f2))                # shear stress at point D\n",
      "print \"The direct stres at point D\", sx1, \"Pa\"\n",
      "print \"The shear stress at point D\", tx1y1, \"Pa\"\n",
      "\n",
      "# Point D'  \n",
      "sx2 = savg + R*math.cos(math.radians(f2))           # Direct stress at point D' \n",
      "tx2y2 = -R*math.sin(math.radians(f2))               # shear stress at point D'\n",
      "print \"The direct stres at point D_desh\", sx2, \"Pa\"\n",
      "print \"The shear stress at point D_desh\", tx2y2, \"Pa\"\n",
      "\n",
      "#Part (b)\n",
      "sp1 =  savg + R                                         # Maximum direct stress in mohe circle (at point P1)\n",
      "tp1 =(f1+180)/2                                         # Inclination of plane of maximum direct stress\n",
      "print \"with angle\", round(tp1,2), \"degree\", \"The maximum direct stress at P1 is \",  sp1, \"Pa\" \n",
      "sp2 =  savg - R                                         # Minimum direct stress in mohe circle (at point P2)\n",
      "tp2 = f1/2                                              # Inclination of plane of minimum direct stress\n",
      "print \"with angle\", round(tp2,2), \"degree\", \"The maximum direct stress at P2 is \", sp2, \"Pa\"\n",
      "\n",
      "# Part (c)\n",
      "tmax = R                                                # Maximum shear stress in mohe circle\n",
      "ts1 = (90 + f1)/2                                       # Inclination of plane of maximum shear stress\n",
      "print \"with plane incilation of\", round(ts1,2), \"degree\", \"The Maximum shear stress is \", tmax, \"Pa\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The direct stres at point D -60000000.0 Pa\n",
        "The shear stress at point D 30000000.0 Pa\n",
        "The direct stres at point D_desh 20000000.0 Pa\n",
        "The shear stress at point D_desh -30000000.0 Pa\n",
        "with angle 116.57 degree The maximum direct stress at P1 is  30000000.0 Pa\n",
        "with angle 26.57 degree The maximum direct stress at P2 is  -70000000.0 Pa\n",
        "with plane incilation of 71.57 degree The Maximum shear stress is  50000000.0 Pa\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 7.7, page no. 520"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"\"\"\n",
      "calculate various quantities\n",
      "\"\"\"\n",
      "\n",
      "import math\n",
      "import numpy\n",
      "\n",
      "#initialisation\n",
      "\n",
      "ex = 340e-06                            # Strain in x-direction\n",
      "ey = 110e-06                            # Strain in y-direction\n",
      "txy = 180e-06                           # shear strain\n",
      "\n",
      "\n",
      "# Part (a)\n",
      "t = 30                                      # Inclination of the element in degree\n",
      "ex1 = (ex+ey)/2.0 + ((ex-ey)/2.0)*math.cos(math.radians(2*t)) + (txy/2.0)*(math.sin(math.radians(2*t)))         # Strain in x1 direction (located at 30 degree)\n",
      "tx1y1 = 2*(-((ex-ey)/2.0)*math.sin(math.radians(2*t)) + (txy/2.0)*(math.cos(math.radians(2*t))))                # Shear starin\n",
      "ey1 = ex+ey-ex1                             # Strain in y1 direction (located at 30 degree)\n",
      "print \"Strain in x1 direction (located at 30 degree) is\", round((ex1/1E-6),2),\"* 10^-6\"\n",
      "print \"shear strain is\", round((tx1y1/1E-6),2),\"* 10^-6\"\n",
      "print \"Strain in y1 direction (located at 30 degree) is\", ey1\n",
      "\n",
      "# Part (b)\n",
      "e1 = (ex+ey)/2.0 + math.sqrt(((ex-ey)/2.0)**2 + (txy/2.0)**2) # Principle stress\n",
      "e2 = (ex+ey)/2.0 - math.sqrt(((ex-ey)/2.0)**2 + (txy/2.0)**2) # Principle stress\n",
      "tp1 = (0.5)*numpy.degrees(numpy.arctan((txy/(ex-ey)))) # Angle to principle stress direction\n",
      "tp2 = 90 + tp1  # Angle to principle stress direction\n",
      "e1 = (ex+ey)/2.0 + ((ex-ey)/2.0)*math.cos(math.radians(2*tp1)) + (txy/2.0)*(math.sin(math.radians(2*tp1)))      # Principle stress via another method\n",
      "e2 = (ex+ey)/2.0 + ((ex-ey)/2.0)*math.cos(math.radians(2*tp2)) + (txy/2.0)*(math.sin(math.radians(2*tp2)))      # Principle stress via another method\n",
      "print \"with angle\", round(tp1,2), \"degree\",\"The Principle stress is \", round(e1,4)\n",
      "print \"with angle\",round(tp2,2), \"degree\",\"The Principle stress is \",round(e2,4)\n",
      "\n",
      "# Part (c)\n",
      "tmax = 2*math.sqrt(((ex-ey)/2.0)**2 + (txy/2.0)**2)         # Maxmum shear strain\n",
      "ts = tp1 + 45                                               # Orientation of element having maximum shear stress \n",
      "tx1y1_ =  2*( -((ex-ey)/2)*math.sin(math.radians(2*ts)) + (txy/2)*(math.cos(math.radians(2*ts))))  # Shear starin assosiated with ts direction\n",
      "print \"with angle\",round(ts,2), \"degree\",\"The Maximum shear strain is \",round(tx1y1_,4)\n",
      "eavg = (e1+e2)/2.0                                          # Average atrain\n",
      "print \"The average strain is\", eavg"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Strain in x1 direction (located at 30 degree) is 360.44 * 10^-6\n",
        "shear strain is -109.19 * 10^-6\n",
        "Strain in y1 direction (located at 30 degree) is 8.95577136594e-05\n",
        "with angle 19.02 degree The Principle stress is  0.0004\n",
        "with angle 109.02 degree The Principle stress is  0.0001\n",
        "with angle 64.02 degree The Maximum shear strain is  -0.0003\n",
        "The average strain is 0.000225\n"
       ]
      }
     ],
     "prompt_number": 5
    }
   ],
   "metadata": {}
  }
 ]
}