{ "metadata": { "name": "chapter 05.ipynb" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter 5:Deflections Of Beams By Double Integration Methods" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.2,Page No.192" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "\n", "#Initilization of Variables\n", "\n", "L=3000 #mm #span of beam\n", "a=2000 #mm\n", "W1=20*10**3 #N #Pt Load Acting on beam\n", "W2=30*10**3 #N #Pt Load Acting on beam\n", "E=2*10**5 #N/mm**2 #Young's Modulus\n", "I=2*10**8 #mm**4 #M.I\n", "\n", "#Calculations\n", "\n", "#Deflection at free End Due to W2\n", "dell1=W2*L**3*(3*E*I)**-1 #mm\n", "\n", "#Deflection at free end Due to W1\n", "dell2=W1*a**3*(3*E*I)**-1+(L-a)*W1*a**2*(2*E*I)**-1 #mm\n", "\n", "#Total Deflection at free end\n", "dell=dell1+dell2 #mm\n", "\n", "#Result\n", "print\"Deflection at Free End is\",round(dell,2),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Deflection at Free End is 9.08 mm\n" ] } ], "prompt_number": 8 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.4,Page No.193" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "\n", "#Initilization of Variables\n", "\n", "E=2*10**5 #N/mm**2 #Young's Modulus\n", "I=180*10**6 #mm**4 #M.I\n", "W1=20 #N/m #u.d.l\n", "W2=20*10**3 #N #Pt load\n", "L=3000 #m #Span of beam\n", "a=2000 #m #Span of u.d.l\n", "\n", "#Calculations\n", "\n", "#Displacement of free End due to 20 KN Pt load at free end\n", "dell1=W2*L**3*(3*E*I)**-1 #mm\n", "\n", "#Displacement of free end due to u.d.l\n", "dell2=W1*a**4*(8*E*I)**-1+(L-a)*W1*a**3*(6*E*I)**-1\n", "\n", "#Deflection at free end\n", "dell=dell1+dell2 #mm\n", "\n", "#Result\n", "print\"The Displacement of Free End of cantilever beam is\",round(dell,2),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The Displacement of Free End of cantilever beam is 6.85 mm\n" ] } ], "prompt_number": 7 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.10,Page No.201" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "\n", "#Initilization of Variables\n", "\n", "E=200*10**6 #KN/m**2 #Young's Modulus\n", "I=15*10**-6 #m**4 #M.I\n", "a=4000 #m \n", "L_AB=6 #m #Span of beam\n", "L_CB=2 #m #Length of CB\n", "F_C=18 #KN #force at C\n", "\n", "#Calculations\n", "\n", "#Let V_A & V_B be the Reactions at A & B Respectively\n", "#V_A+V_B=18\n", "#Now taking moment at B,we get M_B\n", "V_A=(F_C*L_CB)*L_AB**-1\n", "V_B=18-V_A\n", "\n", "#Now Taking Moment at distance x\n", "#M_x=6*x-18*(x-4)\n", "#EI*d**2*y*(d*x**2)**-1=6*x-18*(x-4)\n", "\n", "#Now Integrating above equation,we get\n", "#EI*dy*(dx)**-1=C1+3*x**2-9(x-4)**4\n", "\n", "#Again Integrating above equation we get\n", "#EI*y=C2+C1*x+x**3-3*(x-4)**3\n", "\n", "#The Boundary conditions\n", "x=0\n", "y=0 #.....(a)\n", "\n", "x=6\n", "y=0 #....(b)\n", "\n", "#From Boundary Condition(B.C) a we get\n", "C2=0\n", "\n", "#From Boundary Condition(B.C) b we get\n", "#6*C1+216-3*8\n", "#After Further simplifying we get\n", "C1=-(216-24)*6**-1\n", "\n", "#EI*y=-32*x+x**3-3*(x-4)**3\n", "#EI*dy*(dx)**-1=-32+3*x**2-9(x-4)**4\n", "\n", "#For Max Deflection\n", "#Assume it inthe Porion AC i.e x=4=a\n", "#0=-32+3*x**2\n", "x=(32*3**-1)**0.5\n", "\n", "#Value of Max deflection is\n", "ymax=(-32*x+x**3)*(E*I)**-1 #mm\n", "\n", "#slope at mid-span\n", "\n", "#EI*(dy*(dx)**-1)_centre=-32+3*x**2\n", "#at centre ,\n", "x1=3 #m\n", "\n", "#Let (dy*(dx)**-1)_centre=X\n", "X=-(-32+3*x1**2)*(E*I)**-1 #Radian\n", "\n", "#Deflection at Load Point\n", "x2=4 #m\n", "#EI*y_c=-32*x2+x2**3\n", "\n", "y_c=-(-32*x2+x2**3)*(E*I)**-1\n", "\n", "\n", "#Result\n", "print\"Value of Max Deflection\",round(ymax,4),\"mm\"\n", "print\"SLope at mid-span\",round(X,4),\"radian\"\n", "print\"Deflection at the Load Point is\",round(y_c,4),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Value of Max Deflection -0.0232 mm\n", "SLope at mid-span 0.0017 radian\n", "Deflection at the Load Point is 0.0213 mm\n" ] } ], "prompt_number": 25 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.11,Page No.203" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "\n", "#Initilization of Variables\n", "\n", "L_CB=2 #m #Length of CB\n", "L_AC=4 #m #Length of AB\n", "M_C=15 #KN.m #Moment At Pt C\n", "F_C=30 #KN\n", "L=6 #m Span of Beam\n", "\n", "#Let X=E*I\n", "X=10000 #KN-m**2\n", "\n", "#Calculations\n", "\n", "#Let V_A and V_B be the reactions at A & B respectively\n", "#V_A+V_B=30\n", "\n", "#Taking Moment a A,we get\n", "V_B=(F_C*L_AC+M_C)*L**-1\n", "V_A=30-V_B\n", "\n", "#Now Taking Moment at distacnce x from A\n", "#M_x=7.5*x-30*(x-4)+15\n", "\n", "#By using Macaulay's Method\n", "#EI*(d**2*x/dx**2)=M_x=7.5*x-30*(x-4)+15\n", "\n", "#Now Integrating above Equation we get\n", "#EI*(dy/dx)=C1+7.5*x**2*2**-1-15*(x-4)**2+15*(x-4) ............(1)\n", "\n", "#Again Integrating above Equation we get\n", "#EIy=C2+C1*x+7.5*6**-1*x**3-5*(x-4)**3+15*(x-4)**2*2**-1..........(2)\n", "\n", "#Boundary Cinditions\n", "x=0\n", "y=0\n", "\n", "#Substituting above equations we get \n", "C2=0\n", "\n", "x=6 #m\n", "y=0\n", "\n", "C1=-(7.5*6**3*6**-1-5*2**3+15*2**2*2**-1)*6**-1\n", "\n", "#EIy_c=C2+C1*x+7.5*6**-1*x**3-5*(x-4)**3+15*(x-4)**2*2**-1\n", "#Sub values in Above equation we get\n", "y_c=(93.3333*(X)**-1)\n", "\n", "#Result\n", "print\"The Deflection at C\",round(y_c,4),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The Deflection at C 0.0093 mm\n" ] } ], "prompt_number": 10 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.12,Page No.204" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "\n", "#Initilization of Variables\n", "\n", "L_AC=L_CD=L_DB=2 #m #Length of AC,CD,DB\n", "F_C=40 #KN #Force at C\n", "w=20 #KN/m #u.d.l\n", "L=6 #m #span of beam\n", "\n", "#Let E*I=X\n", "X=15000 #KN-m**2\n", "\n", "\n", "#Calculations\n", "\n", "#Let V_A & V_B be the reactions at A & B respectively\n", "#V_A+V_B=80\n", "\n", "#Taking Moment B,M_B\n", "V_A=(F_C*(L_CD+L_DB)+w*L_DB*L_DB*2**-1)*L**-1 #KN\n", "V_B=80-V_A #KN\n", "\n", "#Taking Moment at distance x from A\n", "#M_x=33.333*x-40*(x-2)-20*(x-4)**2*2**-1\n", "#EI*(d**2/dx**2)=33.333*x-40*(x-2)-10*(x-4)**2\n", "\n", "#Integrating above equation we get\n", "#EI*(dy/dx)=C1+33.333*x**2*2**-1-20*(x-2)**2-10*3**-1*(x-4)**3\n", "\n", "#Again Integrating above equation we get\n", "#EI*y=C2+C1*x+33.333*x**3*6**-1-20*3**-1*(x-2)**3-10*12**-1*(x-4)**4\n", "\n", "#At\n", "x=0\n", "y=0\n", "C2=0\n", "\n", "#At\n", "x=6\n", "y=0\n", "C1=-760*6**-1\n", "\n", "#Assuming Deflection to be max in portion CD and sustituting value of C1 in equation of slope we get\n", "#EI*y=C2+C1*x+33.333*x**3*6**-1-20*3**-1*(x-2)**3-10*12**-1*(x-4)**4\n", "#0=-126.667+33.333*x**2**-1-20*(x-2)**2\n", "\n", "#After rearranging and simplifying further we get\n", "\n", "#x**2-24*x+62=0\n", "#From above equations\n", "a=1\n", "b=-24\n", "c=62\n", "\n", "y=(b**2-4*a*c)**0.5\n", "\n", "x1=(-b+y)*(2*a)**-1\n", "x2=(-b-y)*(2*a)**-1\n", "\n", "#Taking x2 into account\n", "x=2.945 #m\n", "C1=-126.667\n", "C2=0\n", "\n", "y_max=(C2+C1*x+33.333*x**3*6**-1-20*3**-1*(x-2)**3)*X**-1 #mm\n", "\n", "#Max slope occurs at the ends\n", "#At A,\n", "#EI*(dy/dx)_A=-126.667\n", "#At B\n", "#EI*(dy/dx)_B=126.667+33.333*6**2*2**-1-20*4**2-10*2**3\n", "#After simplifying Further we get\n", "#EI*(dy/dx)_B=73.3273\n", "\n", "#Now Max slope is EI(dy/dx)_A=-126.667\n", "#15000*(dy/dx)_=-126.667\n", "\n", "#Let Y=dy/dx\n", "Y=-126.667*X**-1 #Radians\n", "\n", "#Result\n", "print\"Maximum Deflection for Beam is\",round(y_max,4),\"mm\"\n", "print\"Maximum Slope for beam is\",round(Y,4),\"radians\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Maximum Deflection for Beam is -0.0158 mm\n", "Maximum Slope for beam is -0.0084 radians\n" ] } ], "prompt_number": 21 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.13,Page No.206" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "#Initilization of Variables\n", "\n", "E=2*10**8 #KN/m**2\n", "I=450*10**-6 #m**4\n", "L_AC=1 #m #Length of AC\n", "L_CD=3 #m #Length of CD\n", "L_DB=2 #m #Length of DB\n", "w=10 #KN/m #u.d.l\n", "\n", "#Calculations\n", "\n", "#Let V_A & V_B be the reactions at A & B respectively\n", "#V_A+V_B=30\n", "\n", "#Taking Moment at distance x from A\n", "#M_x=17.5*x-10*(x-1)**2*2**-1+10*(x-4)**2*2**-1\n", "#EI*(d**2/dx**2)=17.5*x-10*(x-1)**2*2**-1+10*(x-4)**2*2**-1\n", "\n", "#Now Integrating Above equation we get\n", "#EI(dy/dx)=C1+17.5*x**2*2**-1-5*3**-1*(x-1)**2+5*3**-1*(x-4)**3\n", "\n", "#Again Integrating Above equation we get\n", "#EI*y=C2+C1*x+17.5*x**3*6**-1-5*12**-1*(x-1)**4+5*12**-1*(x-4)**4\n", "\n", "#At \n", "x=0\n", "y=0\n", "C2=0\n", "\n", "#At \n", "x=6 \n", "y=0\n", "C1=(-17.5*x**3*6**-1+5*12**-1*(x-1)**4-5*12**-1*(x-4)**4)*x**-1\n", "\n", "# 1)Slope at A .i.e at x=0\n", "#EI*(dy/dx)_A=C1=-62.708 #KN-m**2\n", "#let (dy/dx)=X\n", "X=C1*(E*I)**-1 #radiams\n", "\n", "#Deflection at mid-span\n", "x=3 #m\n", "#EI*y_centre=C1*x+17.5*x**3*6**-1-5*12**-1*(x-1)**2\n", "y_centre=-(C1*x+17.5*x**3*6**-1-5*12**-1*(x-1)**4)*(E*I)**-1\n", "\n", "#Maximum Deflection\n", "\n", "#At point of Max deflection (dy/dx)=0\n", "#Assuming it in portion CD\n", "\n", "#0=C1*x+17.5*x**2*2**-1-5*3**-1*(x-1)**3\n", "\n", "#Now Let\n", "#F(x)=C1+17.5*x**2*2**-1-5*3**-1*(x-1)**3\n", "\n", "#Let F(x)=Y\n", "#At \n", "x=2.5\n", "Y1=-(C1+17.5*x**2*2**-1-5*3**-1*(x-1)**3)\n", "\n", "#AT\n", "x=3\n", "Y2=-(C1+17.5*x**2*2**-1-5*3**-1*(x-1)**3)\n", "\n", "#At\n", "x=2.9 #m\n", "Y3=-(C1+17.5*x**2*2**-1-5*3**-1*(x-1)**3)\n", "\n", "#A curve may be plotted for (F(x) and the value for which F(x)=0 may be found\n", "#For F(x)=0 for x=2.92 m\n", "#Therefore y_max occur at x=2.92\n", "\n", "x=2.92 #m\n", "y_max=(C1*x+17.5*x**3*6**-1-5*12**-1*(x-1)**4)*(E*I)**-1\n", "\n", "#Result\n", "print\"Slope at A\",round(X,6),\"mm\"\n", "print\"Deflection at mid-span\",round(y_centre,6),\"mm\"\n", "print\"Maxmimum Deflection is\",round(y_max,5),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Slope at A -0.000697 mm\n", "Deflection at mid-span 0.001289 mm\n", "Maxmimum Deflection is -0.00129 mm\n" ] } ], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.14,Page No.208" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "#Initilization of Variables\n", "\n", "L_AC=LDE=L_EB=1 #m #Length of AC\n", "L_CD=2 #m #Length of CD\n", "E=200 #KN/mm**2\n", "I=60*10**6 #mm**4 #M.I\n", "F_C=20 #KN #Force at C\n", "F_E=30 #KN #Force at E\n", "w=10 #KN/m #u.d.l\n", "\n", "#Calculations\n", "\n", "X=E*I*10**-6 #KN-m**2\n", "\n", "#Let V_A & V_B be the reactions at A & B respectively\n", "#V_A+V_B=70\n", "\n", "#Taking Moment at distance x from A\n", "#M_x=34*x-20*(x-1)-10*(x-1)**2*2**-1+10*(x-3)**2*2**-1-30*(x-4)\n", "#EI*(d**2y/dx**2)=34*x-20*(x-1)-10*(x-1)**2*2**-1+10*(x-3)**2*2**-1-30*(x-4)\n", "\n", "#Now Integrating Above equation,we get\n", "#EI*(dy/dx)=C1+17*x**2-10*(x-1)**2-5*3**-1*(x-1)**3+5*3**-1*(x-3)**3-15*(x-4)**2\n", "\n", "#Again Integrating Above equation,we get\n", "#EI*y=C2+C1*x+17*3**-1*x**3-10*3**-1*(x-1)**3-5*12**-1*(x-1)**4+5*12**-1*(x-3)**4-5*(x-4)**3\n", "\n", "#At\n", "x=0\n", "y=0\n", "C2=0\n", "\n", "#At \n", "x=5 #m\n", "y=0\n", "C1=(-17*3**-1*x**3+10*3**-1*(x-1)**3+5*12**-1*(x-1)**4-5*12**-1*(x-3)**4+5*(x-4)**3)*5**-1\n", "\n", "#EI*y=C2+C1*x+17*3**-1*x**3-10*3**-1*(x-1)**3-5*12**-1*(x-1)**4+5*12**-1*(x-3)**4-5*(x-4)**3\n", "C2=0\n", "C1=-78\n", "x=1\n", "y_c=(-78*x+17*3**-1*x)*(X)**-1\n", "\n", "#EI*y_D=C2+C1*x+17*3**-1*x**3-10*3**-1*(x-1)**3-5*12**-1*(x-1)**4\n", "x=3\n", "C1-78\n", "C2=0\n", "y_D=(C2+C1*x+17*3**-1*x**3-10*3**-1*(x-1)**3-5*12**-1*(x-1)**4)*(X**-1)\n", "\n", "#EI*y_E=C2+C1*x+17*3**-1*x**3-10*3**-1*(x-1)**3-5*12**-1*(x-1)**4+5*12**-1*(x-3)**4\n", "x=4\n", "C1-78\n", "C2=0\n", "y_E=(C2+C1*x+17*3**-1*x**3-10*3**-1*(x-1)**3-5*12**-1*(x-1)**4+5*12**-1*(x-3)**4)*X**-1\n", "\n", "#Result\n", "print\"Deflections at C\",round(y_c,5),\"mm\"\n", "print\"Deflections at D\",round(y_D,5),\"mm\"\n", "print\"Deflections at E\",round(y_E,4),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Deflections at C -0.00603 mm\n", "Deflections at D -0.00953 mm\n", "Deflections at E -0.0061 mm\n" ] } ], "prompt_number": 45 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.15,Page No.209" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "%matplotlib inline\n", "\n", "#Initilization of Variables\n", "\n", "E=200 #KN/mm**2 #Modulus of Elasticity\n", "I=300*10**6 #mm\n", "L_AB=L_BC=L_CD=L_DE=1 #m #Length of AB,BC,CD,DE respectively\n", "F_A=20 #KN #Force at A\n", "F_C=10 #KN #Force at C\n", "w=30 #KN/m #u.d.l\n", "\n", "#Let E*I=X\n", "X=E*I*10**-6 #KN-2**2\n", "\n", "#Calculations\n", "\n", "#Let V_E be the reactions at E\n", "V_E=F_A+F_C+w*(L_BC+L_CD) #KN \n", "\n", "#Taking Moment at distance x\n", "#EI*(d**2x/dy**2)=M=-20*x-30*(x-1)**2*2**-1-10*(x-2)+30*(x-3)**2*2**-1\n", "\n", "#Integrating above equation we get\n", "#EI*(dy/dx)=C1-10*x**2-5*(x-1)**3-5*(x-2)**2+5*(x-3)**3\n", "\n", "#Again Integrating above equation\n", "#EI*y=C2+C1*x-10*x**3*3**-1-5*(x-1)**4*4**-1-5*(x-3)**4*4**-1-5*3*(x-2)**3\n", "\n", "#At\n", "#dy/dx=0\n", "x=4 #m\n", "C1=10*x**2+5*(x-1)**3+5*(x-2)**2-5*(x-3)**3\n", "\n", "#AT\n", "x=4\n", "y=0\n", "C2=-C1*4+10*x**3*3**-1+5*(x-1)**4*4**-1-5*(x-3)**4*4**-1+5*3**-1*(x-2)**3\n", "\n", "#Max Deflection and Max slopes occurs at Free end in case of cantilever\n", "y_max=y_A=C2*X**-1\n", "\n", "#EI*(dy/dx)_max=C1\n", "#Let (dy/dx)=Y\n", "Y=C1*X**-1 #radian\n", "\n", "#Now deflection at x=1 #m\n", "C2=-913.333\n", "C1=310\n", "x=1\n", "y_B=(C2+C1*x-10*x**3*3**-1)*X**-1\n", "\n", "#Now Deflection at x=2 #m\n", "C2=-913.333\n", "C1=310\n", "x=2 #m\n", "y_C=(C2+C1*x-10*x**3*3**-1-5*(x-1)**4*4**-1)*X**-1\n", "\n", "#Now Deflection at x=3 #m\n", "C2=-913.333\n", "C1=310\n", "x=3 #m\n", "y_D=(C2+C1*x-10*x**3*3**-1-5*(x-1)**4*4**-1-5*3**-1*(x-2)**3)*X**-1\n", "\n", "y_E=0\n", "\n", "#Result\n", "print\"Max Deflection for Beam\",round(y_A,4),\"mm\"\n", "print\"Max Slope for beam\",round(Y,5),\"radians\"\n", "\n", "#Plotting the ELastic Curve\n", "\n", "Y2=[y_E,y_D,y_C,y_B,y_A]\n", "X2=[L_AB+L_BC+L_CD+L_DE,L_AB+L_BC+L_CD,L_AB+L_BC,L_AB,0]\n", "Z2=[0,0,0,0,0]\n", "plt.plot(X2,Y2,X2,Z2)\n", "plt.xlabel(\"Length in mm\")\n", "plt.ylabel(\"Deflection in mm\")\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Max Deflection for Beam -0.0152 mm\n", "Max Slope for beam 0.00517 radians\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.16,Page No.211" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "#Initilization of Variables\n", "\n", "L_BD=L_CB=L_AC=2 #m #Length of BD,CB,AC\n", "F_C=40 #KN #Force at C\n", "F_D=10 #KN Force at D\n", "L=6 #m spna of beam\n", "\n", "#EI is constant in this problem\n", "\n", "#Calculations\n", "\n", "#Let V_A & V_B be the reactions at A & B Respectively\n", "#V_A+V_B=50\n", "\n", "#Taking Moment at Pt A\n", "V_B=(F_D*L+F_C*L_AC)*(L_AC+L_CB)**-1\n", "V_A=50-V_B\n", "\n", "#Now Taking Moment at distance x from A,M_x\n", "#M_x=15*x-40*(x-2)+35*(x-4)\n", "#EI*(d**2*y/dx**2)=15*x-40*(x-2)+35*(x-4)\n", "\n", "#Now Integrating above equation we get\n", "#EI*(dy/dx)=C1+7.5*x**2-20*(x-2)**2+17.5(x-4)**2\n", "\n", "#Again Integrating above equation we get\n", "#EI*y=C2+C1*x+2.5*x**2-20*3**-1*(x-2)**3+17.5*(x-4)**3*3**-1\n", "\n", "#At\n", "x=0\n", "y=0\n", "#we get\n", "C2=0\n", "\n", "#At\n", "x=4 \n", "y=0\n", "#we get\n", "C1=(2.5*4**3-20*3**-1*2**3)*4**-1\n", "\n", "#Now Deflection at C\n", "x=2\n", "C1=-26.667\n", "C2=0\n", "y_C=C2+C1*x+2.5*x**3\n", "\n", "#Now Deflection at D\n", "C1=-21.667\n", "C2=0\n", "y_D=-26.667*6+2.5*6**3-20*3**-1*4**3+17.5*2**3*3**-1\n", "\n", "#Result\n", "print\"Deflections Under Loads are:y_D\",round(y_D,4)\n", "print\" :y_C\",round(y_C,2)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Deflections Under Loads are:y_D -0.002\n", " :y_C -33.33\n" ] } ], "prompt_number": 8 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.17,Page No.212" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "#Initilization of Variables\n", "\n", "L_BC=L_EB=2 #m #Length of BC & EB\n", "E=200*10**6 #KN/m**2 #Modulus of eLasticity\n", "I=45*10**-6 #mm**4 #M.I\n", "L_DE=3 #m #Length of DE\n", "L_AD=1 #m #Length of AD\n", "w=20 #KN/m #u.d.l\n", "L=8 #m #span of beam\n", "F_C=30 #KN #Force at C\n", "\n", "#Calculations\n", "\n", "#Let V_A & V_B be the reactions at A & B respectively\n", "#V_A+V_B=90\n", "\n", "#Taking Moment at A,M_A\n", "V_B=(w*L_DE*(L_DE*2**-1+L_AD)+F_C*L)*(L_AD+L_DE+L_EB)**-1\n", "V_A=90-V_B\n", "\n", "#Taking Moment at distance x\n", "#M_x=25*x-20*(x-1)**2*2**-1+20*(x-4)**2*2**-1+65*(x-6)\n", "\n", "#Integrating above equation we get\n", "#EI*(d**2*y/dx**2)=25*x-10*(x-1)**2+10*(x-4)**2+65*(x-6)\n", "\n", "#again Integrating above equation we get\n", "#EI*(dy/dx)=C1+25*x**2*2**-1-10*3**-1*(x-1)**3+10*3**-1*(x-4)**2+65*2**-1*(x-6)**2\n", "\n", "#again Integrating above equation we get\n", "#EI*y=C2+C1*x+25*6**-1*x**3-10*12**-1*(x-1)**4+10*12**-1*(x-4)**4+65*6**-1*(x-6)**3\n", "\n", "x=0\n", "y=0\n", "#Sub these values in above equation,we get\n", "C2=0\n", "\n", "x=6 #m\n", "y=0\n", "C1=-(25*6**-1*6**3-10*12**-1*5**4+10*12**-1*2**4)*6**-1\n", "\n", "#deflection at C is given by\n", "x=8\n", "y_c=(C2+C1*x+25*6**-1*x**3-10*12**-1*(x-1)**4+10*12**-1*(x-4)**4+65*6**-1*(x-6)**3)*(E*I)**-1\n", "\n", "#Assuming y is max in the portion DE,then\n", "#(dy/dx)=0 for that point\n", "\n", "#0=-65.417+25*2**-1*x**2-10*3**-1*x(-1)**3\n", "\n", "#Let F(x)=-65.417+25*2**-1*x**2-10*3**-1*x(-1)**3\n", "#Let z=F(x)\n", "\n", "#AT \n", "x=3\n", "z=-65.417+25*2**-1*x**2-10*3**-1*(x-1)**3\n", "\n", "x=2.5\n", "z1=-65.417+25*2**-1*x**2-10*3**-1*(x-1)**3\n", "\n", "x=2.4\n", "z2=-65.417+25*2**-1*x**2-10*3**-1*(x-1)**3\n", "\n", "#The assumption is max in portion DE\n", "x=2.46\n", "y_max=(-65.417*x+25*6**-1*x**3-10*12**-1*1.46**4)*(E*I)**-1\n", "\n", "#Result\n", "print\"Deflection at free end C\",round(y_c,4),\"mm\"\n", "print\"Max Deflection between A and B\",round(y_max,4),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Deflection at free end C -0.0101 mm\n", "Max Deflection between A and B -0.0114 mm\n" ] } ], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 5.5.18,Page No.213" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import numpy as np\n", "\n", "#Initilization of Variables\n", "\n", "L_DB=L_AC=L_ED=2 #m #Length of DB & AC\n", "L_CD=4 #m #Length of CD\n", "L_CE=2 #m #Length of CE\n", "F_A=40 #KN #Force at C\n", "F_B=20 #KN #Force at A\n", "E=200*10**6 #KN/mm**2 #Modulus of Elasticity\n", "I=50*10**-6 #m**4 #M.I\n", "\n", "#Calculations\n", "\n", "#LEt V_C & V_D be the reactions at C & D respectively\n", "#V_C+V_D=60\n", "\n", "#Taking Moment At D,M_D\n", "V_C=-(-F_A*(L_AC+L_CE+L_ED)+F_B*L_DB)*L_CD**-1\n", "V_D=60-V_C\n", "\n", "#Now Taking Moment at Distance x from A,\n", "#M_x=-40*x+50*(x-2)+10*(x-6)\n", "\n", "#EI*(d**2*y/dx**2)=-40*x+50*(x-2)+10*(x-6)\n", "\n", "#Now Integrating above Equation we get\n", "#EI*(dy/dx)=C1+20*x**2-25*(x-2)+5*(x-6)**2\n", "\n", "#Again Integrating above Equation we get\n", "#EI*y=C2+C1*x-20*3**-1*x**3+25*3**-1*(x-2)**3+5*3**-1*(x-6)**3\n", "\n", "#At\n", "x=0\n", "y=0\n", "#C2+2*C1=-53.33 ...............(1)\n", "\n", "#At \n", "x=6\n", "y=0\n", "#C2+6*C1=906.667 ...............(2)\n", "\n", "#Subtracting Equation 1 from 2 we get\n", "C1=853.333*4**-1\n", "C2=53.333-2*C1\n", "x=0\n", "y_A=(C2+C1*x-20*3**-1*x**3+25*3**-1*(x-2)**3+5*3**-1*(x-6)**3)*(E*I)**-1\n", "\n", "#Answer For y_A is incorrect in textbook\n", "\n", "#At Mid-span\n", "C1=853.333*4**-1\n", "C2=53.333-2*C1\n", "x=4\n", "y_E=(C2+C1*x-20*3**-1*x**3+25*3**-1*(x-2)**3+5*3**-1*(x-6)**3)*(E*I)**-1\n", "\n", "#Answer For y_E is incorrect in textbook\n", "\n", "#At B\n", "C1=853.333*4**-1\n", "C2=53.333-2*C1\n", "x=8\n", "y_B=(C2+C1*x-20*3**-1*x**3+25*3**-1*(x-2)**3+5*3**-1*(x-6)**3)*(E*I)**-1\n", "\n", "\n", "#Result\n", "print\"Deflection relative to the level of the supports:at End A\",round(y_A,4),\"mm\"\n", "print\" :at End B\",round(y_B,4),\"mm\"\n", "print\" :at Centre of CD\",round(y_E,4),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Deflection relative to the level of the supports:at End A -0.08 mm\n", " :at End B -0.0267 mm\n", " :at Centre of CD 0.0107 mm\n" ] } ], "prompt_number": 8 } ], "metadata": {} } ] }