path: root/Mechanics_of_Materials_by_Pytel_and_Kiusalaas/Chapter06.ipynb

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 06:Deflection of Beam"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.1, Page No:196"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "wo=400 #loading in lb/ft\n",
      "E=29*10**6 #Modulus of elasticity in psi\n",
      "I=285 #Moment of inertia in in^4\n",
      "S=45.6 #Sectional Modulus in in^3\n",
      "L=8 #Span in ft\n",
      "\n",
      "#Calculations\n",
      "#Part 1\n",
      "#Part1 is theoretical in nature hence not coded\n",
      "\n",
      "#Part 2\n",
      "delta_max=((wo*12**-1)*(L*12)**4)/(8*E*I) #maximum deflection in inches\n",
      "M_max=(wo*12**-1)*(L*12)**2 #Maximum moment\n",
      "sigma_max=M_max/(2*S) #Maximum bending stress in psi\n",
      "\n",
      "#Result\n",
      "print M_max\n",
      "print \"The maximum deflection is\",round(delta_max,4),\"in\"\n",
      "print \"The maximum Bending Stress is\",round(sigma_max),\"psi\"\n",
      "\n",
      "#Answer in the textbook for sigma_max is incorrect"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "307200.0\n",
        "The maximum deflection is 0.0428 in\n",
        "The maximum Bending Stress is 3368.0 psi\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.3, Page No:198"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "P=300 #Point Load in N\n",
      "R_a=100 #Reaction at A in N\n",
      "R_c=200 #Reaction at C in N\n",
      "E=12 #Youngs Modulus in GPa\n",
      "L1=2 #Length of the load from A in m\n",
      "L2=1 #Length of the load from C in m\n",
      "b=0.04 #Width of the CS of the beam in m\n",
      "h=0.08 #Depth of the CS of the beam in m\n",
      "\n",
      "#Claculations\n",
      "#Moment of inertia \n",
      "I=b*h**3*12**-1 #Moment of Inertia in m^4\n",
      "#Flexural Rigidity\n",
      "FR=E*10**9*I #FLexural rigidity in N.m^2\n",
      "\n",
      "#Moments in terms of x are\n",
      "#Given\n",
      "#After the variable Calculations we get\n",
      "C1=-400/3 #Constant\n",
      "C3=C1 #Constant\n",
      "C2=0 #Constant\n",
      "C4=0 #Constant\n",
      "\n",
      "#to get max displacement x we have\n",
      "x=(6.510/2.441)**0.5 #Length at which displacement is maximum in m\n",
      "v=(0.8138*x**3-6.510*x) #Displacement in mm\n",
      "\n",
      "#Largest slope\n",
      "theta=(2.441*(L1+L2)**2-(7.324*(L1+L2-L1)**2)-6.150)*10**-3#Angle in radians\n",
      "\n",
      "#Result \n",
      "print \"The maximum displacement is\",round(-v,2),\"mm downwards\"\n",
      "print \"The maximum angle is\",round(theta*180*pi**-1,3),\"degrees anticlockwise\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The maximum displacement is 7.09 mm downwards\n",
        "The maximum angle is 0.487 degrees anticlockwise\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.4, Page No:200"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "#The computation is mostly variable based hence values will be directly declared \n",
      "C1=19.20*10**3 #lb.ft^2\n",
      "C2=-131.6*10**3 #lb.ft^2\n",
      "C3=14.7*10**3 #lb.ft^2\n",
      "C4=-112.7*10**3 #lb.ft^2\n",
      "EI=10**7 #Flexural Rigidity in psi\n",
      "\n",
      "#Calculations\n",
      "v=-(C2*12**3)/(EI*40) #Displacement in inches\n",
      "\n",
      "#Result\n",
      "print \"The maximum displacement is\",round(v,3),\"in downwards\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The maximum displacement is 0.569 in downwards\n"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.6, Page No:210"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "L1=3 #Length in m\n",
      "L2=1 #Length in m\n",
      "L3=8 #Length in m\n",
      "L4=4 #Length in m\n",
      "L5=6 #Length in m\n",
      "\n",
      "#Calculations\n",
      "#Deflection midway\n",
      "EIv=250*3**-1*L1**3-(50*3**-1*(L1-L2)**4)-(3925*3**-1*L1) #Deflection in N.m^3\n",
      "\n",
      "#Deflection at E\n",
      "EIv_E=250*3**-1*L3**3-(50*3**-1*(L3-L2)**4)+(50*3**-1*(L3-L4)**4)+(650*3**-1*(L3-L5)**3)-(3925*3**-1*L3) #Deflection in N.m^3\n",
      "\n",
      "#Result\n",
      "print \"The deflection at midspan is\",round(-EIv),\"N.m^3 downwards\"\n",
      "print \"The deflection at point E is\",round(-EIv_E),\"N.m^3 downwards\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The deflection at midspan is 1942.0 N.m^3 downwards\n",
        "The deflection at point E is 1817.0 N.m^3 downwards\n"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.8, Page No:223"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "x1=16*3**-1 #Centroid of the triangle in ft\n",
      "x2=3 #Centroid of the lower parabola in ft\n",
      "x3=6 #Centroid of the rectangle in ft\n",
      "x4=20*3**-1 #Centroid of the triangle in ft\n",
      "#Moment values\n",
      "M1=4800 #Moment in lb.ft\n",
      "M2=14400 #Moment in lb.ft\n",
      "\n",
      "#Calcualtions\n",
      "P=((3**-1*4*M1*x2)+(4*M1*x3)+(0.5*4*M1*2*x4))*(x1*8*8*0.5)**-1 #Force P in lb\n",
      "\n",
      "#Result\n",
      "print \"The magnitude of force P is\",round(P,1),\"lb\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The magnitude of force P is 1537.5 lb\n"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.9, Page No:225"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "P=300 #Force in N\n",
      "L1=1 #Length in m\n",
      "L2=2 #Length in m\n",
      "R_a=100 #Reaction at A in N\n",
      "R_c=200 #Reaction at C in N\n",
      "EI=20.48*10**3 #Flexural Rigidity in N.m^2\n",
      "\n",
      "#Calculations\n",
      "#Part 1\n",
      "tC_A=(0.5*(L1+L2)*P*L1-(0.5*L1*P*(L1+L2)**-1))*EI**-1 #First Moment in m\n",
      "theta_A=tC_A/(L1+L2) #Angle in radians \n",
      "\n",
      "#Part 2\n",
      "tD_A=0.5*L1*R_a*(L1+L2)**-1*EI**-1 #First Moment in m\n",
      "delta_D=(theta_A*L1-tD_A) #Displacement in m \n",
      "\n",
      "#Result\n",
      "print \"The angle in part 1 is\",round(theta_A*180*pi**-1,3),\"Degrees\"\n",
      "print \"The displacement in part 2 is\",round(delta_D*1000,2),\"mm downward\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The angle in part 1 is 0.373 Degrees\n",
        "The displacement in part 2 is 5.7 mm downward\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.10, Page No:227"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "P1=150 #Load in lb\n",
      "P2=30 #Load in lb\n",
      "R_A=78 #Reaction at A in lb\n",
      "R_C=102 #Reaction at C in lb\n",
      "L1=4 #Length in ft\n",
      "L2=6 #Length in ft\n",
      "M1=780 #Moment in lb.ft\n",
      "M2=900 #Moment in lb.ft\n",
      "M3=120 #Moment in lb.ft\n",
      "\n",
      "#Calculations\n",
      "EI_AC=0.5*(L1+L2)*M1*(2*3**-1)*(L1+L2)-(0.5*L2*M2*(L1+(2*3**-1)*L2)) #Deflection in lb.ft^3\n",
      "EI_thetaC=EI_AC/(L1+L2) #Deflection in lb.ft^2\n",
      "\n",
      "EI_DC=-0.5*L1*M3*2*3**-1*L1 #Deflection in lb.ft^3\n",
      "EI_deltaD=EI_thetaC*L1-(-EI_DC) #Deflection in lb.ft^2\n",
      "\n",
      "#Result\n",
      "print \"The deflection is\",round(EI_deltaD),\"lb.ft^2 upwards\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The deflection is 1120.0 lb.ft^2 upwards\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.11, Page No:234"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "P1=80 #Load in lb\n",
      "P2=100 #Load in lb\n",
      "b1=3 #Distance of load from end in ft\n",
      "b2=2 #Distance of load from end in ft\n",
      "L=9 #Span of the beam in ft\n",
      "\n",
      "#Calcualtions\n",
      "EI_delta1=(P1*b1*48**-1)*(3*L**2-4*b1**2) #Deflection in lb.ft^3\n",
      "EI_delta2=(P2*b2*48**-1)*(3*L**2-4*b2**2) #Deflection in lb.ft^3\n",
      "EI_delta=EI_delta1+EI_delta2 #Deflection at modspan in lb.ft^3\n",
      "\n",
      "#Result\n",
      "print \"The deflection at midspan is\",round(EI_delta),\" lb.ft^3 downward\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The deflection at midspan is 1981.0  lb.ft^3 downward\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 6.6.12, Page no:234"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable Decleration\n",
      "wo=600 #Load in N/m\n",
      "L=6 #Span of the beam in m\n",
      "b=2 #Distance of the load from end in m\n",
      "a=1 #Distance of the load from end in m\n",
      "\n",
      "#Calulations\n",
      "EI_delta1=wo*384**-1*(5*L**4-12*L**2*b**2+8*b**4) #Deflection in N.m^3\n",
      "EI_delta2=wo*96**-1*a**2*(3*L**2-2*a**2) #Deflection in N.m^3\n",
      "\n",
      "EI_delta=EI_delta1-EI_delta2 #Total Delfection at midspan in N.m^3\n",
      "\n",
      "#Result\n",
      "print \"The total Deflection at midpsan is\",round(EI_delta),\"N.m^3 downwards\"\n",
      "\n",
      "#NOTE:The answer varies due to decimal point accuracy"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The total Deflection at midpsan is 6963.0 N.m^3 downwards\n"
       ]
      }
     ],
     "prompt_number": 16
    }
   ],
   "metadata": {}
  }
 ]
}