ex 4.9 with additional point load

ex4.9_mod_pub.ipynb

{
 "metadata": {
  "name": "Ex 4-9 modified_and Numerical"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Example 4.9 with slight modification, Added a point force P acting donward at node 2"
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "from sympy import *\nfrom sympy import Matrix\nfrom sympy import init_printing\nfrom IPython.display import Latex, Math, display #allow printing latex inline with formatted matricies\ninit_printing()\n#%load_ext sympy.interactive.ipythonprinting\n\n\n\n\nE = Symbol('E')\nI = Symbol('I')\nL = Symbol('L')\nP2 = Symbol('P2')\nM2 = Symbol('M2')\nM3 = Symbol('M3')\nw = Symbol ('w')\nP= Symbol('P')\nF1y= Symbol('F_{1y}')\nF3y= Symbol('F_{3y}')\nM1= Symbol('M_{1}')\nM3= Symbol('M_{3}')\nD = Symbol('D')\nv2 = Symbol('v2')\nphi2 = Symbol('\\phi_2')\n",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "from IPython.display import HTML\n\nHTML('''<script>\ncode_show=true; \nfunction code_toggle() {\n if (code_show){\n $('div.input').hide();\n } else {\n $('div.input').show();\n }\n code_show = !code_show\n} \n$( document ).ready(code_toggle);\n</script>\nThe raw code for this IPython notebook is by default hidden for easier reading.\nTo toggle on/off the raw code, click <a href=\"javascript:code_toggle()\">here</a>.''')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": "<script>\ncode_show=true; \nfunction code_toggle() {\n if (code_show){\n $('div.input').hide();\n } else {\n $('div.input').show();\n }\n code_show = !code_show\n} \n$( document ).ready(code_toggle);\n</script>\nThe raw code for this IPython notebook is by default hidden for easier reading.\nTo toggle on/off the raw code, click <a href=\"javascript:code_toggle()\">here</a>.",
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": "<IPython.core.display.HTML at 0xa163650>"
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Define element stiffness"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "k= (1/L**3)*E*I*Matrix([[12, 6*L, -12, 6*L], [6*L, 4*L*L, -6*L, 2*L*L], [-12, -6*L, 12, -6*L], [6*L, 2*L*L, -6*L, 4*L*L] ])\ndisplay(Math(latex('[k]= ')+latex(k)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$[k]= \\left[\\begin{smallmatrix}{}\\frac{12 E}{L^{3}} I & \\frac{6 E}{L^{2}} I & - \\frac{12 E}{L^{3}} I & \\frac{6 E}{L^{2}} I\\\\\\frac{6 E}{L^{2}} I & \\frac{4 E}{L} I & - \\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I\\\\- \\frac{12 E}{L^{3}} I & - \\frac{6 E}{L^{2}} I & \\frac{12 E}{L^{3}} I & - \\frac{6 E}{L^{2}} I\\\\\\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I & - \\frac{6 E}{L^{2}} I & \\frac{4 E}{L} I\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x13710fb0>"
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "K_g=zeros(6)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Add element 1 to global matrix"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "K_g[0:4,0:4] = K_g[0:4,0:4] +k\ndisplay(Math(latex('[K]= ')+latex(K_g)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$[K]= \\left[\\begin{smallmatrix}{}\\frac{12 E}{L^{3}} I & \\frac{6 E}{L^{2}} I & - \\frac{12 E}{L^{3}} I & \\frac{6 E}{L^{2}} I & 0 & 0\\\\\\frac{6 E}{L^{2}} I & \\frac{4 E}{L} I & - \\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I & 0 & 0\\\\- \\frac{12 E}{L^{3}} I & - \\frac{6 E}{L^{2}} I & \\frac{12 E}{L^{3}} I & - \\frac{6 E}{L^{2}} I & 0 & 0\\\\\\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I & - \\frac{6 E}{L^{2}} I & \\frac{4 E}{L} I & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163b30>"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Add element 2 to global matrix"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "K_g[2:6,2:6]=K_g[2:6,2:6] + k\ndisplay(Math(latex('[K]= ')+latex(K_g)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$[K]= \\left[\\begin{smallmatrix}{}\\frac{12 E}{L^{3}} I & \\frac{6 E}{L^{2}} I & - \\frac{12 E}{L^{3}} I & \\frac{6 E}{L^{2}} I & 0 & 0\\\\\\frac{6 E}{L^{2}} I & \\frac{4 E}{L} I & - \\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I & 0 & 0\\\\- \\frac{12 E}{L^{3}} I & - \\frac{6 E}{L^{2}} I & \\frac{24 E}{L^{3}} I & 0 & - \\frac{12 E}{L^{3}} I & \\frac{6 E}{L^{2}} I\\\\\\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I & 0 & \\frac{8 E}{L} I & - \\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I\\\\0 & 0 & - \\frac{12 E}{L^{3}} I & - \\frac{6 E}{L^{2}} I & \\frac{12 E}{L^{3}} I & - \\frac{6 E}{L^{2}} I\\\\0 & 0 & \\frac{6 E}{L^{2}} I & \\frac{2 E}{L} I & - \\frac{6 E}{L^{2}} I & \\frac{4 E}{L} I\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163a70>"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Work equivalent nodal loads for element 1"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f1_0 = Matrix([-3*w*L/40, -(w*L**2)/60, -7*w*L/40, (w*L**2)/40])\ndisplay(Math(latex('\\{f^{(1)}_0\\}= ')+latex(f1_0)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{f^{(1)}_0\\}= \\left[\\begin{smallmatrix}{}- \\frac{3 L}{40} w\\\\- \\frac{L^{2} w}{60}\\\\- \\frac{7 L}{40} w\\\\\\frac{L^{2} w}{40}\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163c30>"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Work equivalent nodal loads for element 2"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f2_0 = Matrix([-13*w*L/40, -(7*w*L**2)/120, -17*w*L/40, (w*L**2)/15])\ndisplay(Math(latex('\\{f^{(2)}_0\\}= ')+latex(f2_0)))\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{f^{(2)}_0\\}= \\left[\\begin{smallmatrix}{}- \\frac{13 L}{40} w\\\\- \\frac{7 w}{120} L^{2}\\\\- \\frac{17 L}{40} w\\\\\\frac{L^{2} w}{15}\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163c30>"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "System load array"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "F = zeros(6,1)\nF[0] = F1y\nF[1] = M1\nF[2] = -P\nF[4] = F3y\ndisplay(Math(latex('\\{F\\} = ')+latex(F)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{F\\} = \\left[\\begin{smallmatrix}{}F_{{1y}}\\\\M_{{1}}\\\\- P\\\\0\\\\F_{{3y}}\\\\0\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163b50>"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Convert to numerical values with L = 100in, w = 20 lb/in, P=1000 lb, E = 30E6 psi, I =100 in^4"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "#convert to numerical values with L = 100in, w = 20 lb/in, P=1000 lb, E = 30E6 psi, I =100 in^4\nkn = k.subs({L:100, E:30e6, I:100, w:20, P:1000})\nkn = np.array(np.array(kn), np.float)\n\nK_gn = K_g.subs({L:100, E:30e6, I:100, w:20, P:1000})\nK_gn = np.array(np.array(K_gn), np.float)\n\nf1_0n = f1_0.subs({L:100, E:30e6, I:100, w:20, P:1000})\nf1_0n = np.array(np.array(f1_0n), np.float)\n\nf2_0n = f2_0.subs({L:100, E:30e6, I:100, w:20, P:1000})\nf2_0n = np.array(np.array(f2_0n), np.float)\n\nFn = F.subs({L:100, E:30e6, I:100, w:20, P:1000, F1y:0, F3y:0, M1:0, M3:0})\nFn = np.array(np.array(Fn), np.float)\n\n\n",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Enforce constraints and solve for {d}"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "F_0n = np.zeros((6,1))\nF_0n[0:4] = f1_0n \nF_0n[2:6] = f2_0n +F_0n[2:6]\n\nF_rn = F_0n + Fn\n\ndisplay(Math(latex('\\{F\\} = [K]\\{d\\} - \\{F_0\\} ')))\n\nD = [0, 0, v2, phi2, 0, 0]\n\ndisplay(Math( latex(F) + latex('=') + latex(Matrix(K_gn).evalf(4)) + latex(Matrix(D)) + latex('-')+ latex(Matrix(F_0n).evalf(5))))\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{F\\} = [K]\\{d\\} - \\{F_0\\} $$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x1374b2f0>"
      },
      {
       "latex": "$$\\left[\\begin{smallmatrix}{}F_{{1y}}\\\\M_{{1}}\\\\- P\\\\0\\\\F_{{3y}}\\\\0\\end{smallmatrix}\\right]=\\left[\\begin{smallmatrix}{}3.6 \\cdot 10^{4} & 1.8 \\cdot 10^{6} & -3.6 \\cdot 10^{4} & 1.8 \\cdot 10^{6} & 0 & 0\\\\1.8 \\cdot 10^{6} & 1.2 \\cdot 10^{8} & -1.8 \\cdot 10^{6} & 6.0 \\cdot 10^{7} & 0 & 0\\\\-3.6 \\cdot 10^{4} & -1.8 \\cdot 10^{6} & 7.2 \\cdot 10^{4} & 0 & -3.6 \\cdot 10^{4} & 1.8 \\cdot 10^{6}\\\\1.8 \\cdot 10^{6} & 6.0 \\cdot 10^{7} & 0 & 2.4 \\cdot 10^{8} & -1.8 \\cdot 10^{6} & 6.0 \\cdot 10^{7}\\\\0 & 0 & -3.6 \\cdot 10^{4} & -1.8 \\cdot 10^{6} & 3.6 \\cdot 10^{4} & -1.8 \\cdot 10^{6}\\\\0 & 0 & 1.8 \\cdot 10^{6} & 6.0 \\cdot 10^{7} & -1.8 \\cdot 10^{6} & 1.2 \\cdot 10^{8}\\end{smallmatrix}\\right]\\left[\\begin{smallmatrix}{}0\\\\0\\\\v_{2}\\\\\\phi_{2}\\\\0\\\\0\\end{smallmatrix}\\right]-\\left[\\begin{smallmatrix}{}-150.0\\\\-3333.3\\\\-1000.0\\\\-6666.7\\\\-850.0\\\\13333.0\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa16d050>"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "D_rn = dot(inv(K_gn[2:4,2:4]),F_rn[2:4])\nD_n = np.zeros((6,1))\nD_n[2:4] = D_rn \ndisplay(Math(latex('\\{d\\}= ')+latex(Matrix(D_n).evalf(4))))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{d\\}= \\left[\\begin{smallmatrix}{}0\\\\0\\\\-0.02778\\\\-2.778 \\cdot 10^{-5}\\\\0\\\\0\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163ff0>"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "System force array (Reactions)"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "#reactions\nR_n = dot(K_gn,D_n) - F_0n\ndisplay(Math(latex('\\{F\\}= ')+latex(Matrix(R_n).evalf(5))))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{F\\}= \\left[\\begin{smallmatrix}{}1100.0\\\\51667.0\\\\-1000.0\\\\0\\\\1900.0\\\\-65000.0\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163d50>"
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Element Forces"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "#element forces\nf1n = dot(kn,D_n[0:4]) - f1_0n\ndisplay(Math(latex('\\{f^1\\}= ')+latex(Matrix(f1n).evalf(5))))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{f^1\\}= \\left[\\begin{smallmatrix}{}1100.0\\\\51667.0\\\\-600.0\\\\41667.0\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163e70>"
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f2n = dot(kn,D_n[2:6]) - f2_0n\ndisplay(Math(latex('\\{f^2\\}= ')+latex(Matrix(f2n).evalf(5))))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\{f^2\\}= \\left[\\begin{smallmatrix}{}-400.0\\\\-41667.0\\\\1900.0\\\\-65000.0\\end{smallmatrix}\\right]$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163c30>"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Calculate shear and moment based on differential realtions"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "v1 = Symbol('v1')\nv2 = Symbol('v2')\nphi1 = Symbol('phi1')\nphi2 = Symbol('phi2')\nx = Symbol('x')\nv = ((2/L**3)*(v1-v2) + (1/L**2)*(phi1 +phi2))*x**3 + (-3/(L**2)*(v1-v2) - 1/L*(2*phi1 + phi2))*x**2 + phi1*x + v1\ndisplay(Latex('recall our displacement function:'))\ndisplay(Math(latex(r'\\tilde{v}(x) = ')+latex(v)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "recall our displacement function:",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0x1374b5d0>"
      },
      {
       "latex": "$$\\tilde{v}(x) = \\phi_{1} x + v_{1} + x^{3} \\left(\\frac{1}{L^{2}} \\left(\\phi_{1} + \\phi_{2}\\right) + \\frac{1}{L^{3}} \\left(2 v_{1} - 2 v_{2}\\right)\\right) + x^{2} \\left(- \\frac{1}{L} \\left(2 \\phi_{1} + \\phi_{2}\\right) - \\frac{1}{L^{2}} \\left(3 v_{1} - 3 v_{2}\\right)\\right)$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x1374b5f0>"
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "V=E*I*diff(v,x,3)\ndisplay(Math(latex(r'V(x)=EI\\frac{d^3\\tilde{v}(x)}{dx^3} = ')+latex(V)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$V(x)=EI\\frac{d^3\\tilde{v}(x)}{dx^3} = \\frac{6 E}{L^{2}} I \\left(\\phi_{1} + \\phi_{2} + \\frac{1}{L} \\left(2 v_{1} - 2 v_{2}\\right)\\right)$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x1371c230>"
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "V1= V.subs({L:100, E:30e6, I:100, v1:D_n[0], phi1:D_n[1], v2:D_n[2], phi2:D_n[3] })\ndisplay(Latex('The shear force in element 1:'))\ndisplay(Math(latex('V^{(1)}= ')+latex(V1.evalf(5))))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "The shear force in element 1:",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0x1374b270>"
      },
      {
       "latex": "$$V^{(1)}= 950.0$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0xa163ed0>"
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "V2= V.subs({L:100, E:30e6, I:100, v1:D_n[2], phi1:D_n[3], v2:D_n[4], phi2:D_n[5] })\ndisplay(Latex('The shear force in element 2:'))\ndisplay(Math(latex('V^{(2)}= ')+latex(V2.evalf(5))))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "The shear force in element 2:",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0x1374b450>"
      },
      {
       "latex": "$$V^{(2)}= -1050.0$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x1374b150>"
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "M=E*I*diff(v,x,2)\ndisplay(Latex('The bending moment:'))\ndisplay(Math(latex(r'M(x)=EI\\frac{d^2\\tilde{v}(x)}{dx^2} = ')+latex(M)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "The bending moment:",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0x1374b4b0>"
      },
      {
       "latex": "$$M(x)=EI\\frac{d^2\\tilde{v}(x)}{dx^2} = \\frac{2 E}{L} I \\left(- 2 \\phi_{1} - \\phi_{2} + \\frac{3 x}{L} \\left(\\phi_{1} + \\phi_{2} + \\frac{1}{L} \\left(2 v_{1} - 2 v_{2}\\right)\\right) - \\frac{1}{L} \\left(3 v_{1} - 3 v_{2}\\right)\\right)$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x1374b750>"
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Me1= M.subs({L:100, E:30e6, I:100, v1:D_n[0], phi1:D_n[1], v2:D_n[2], phi2:D_n[3] })\ndisplay(Latex('The bending moment in element 1:'))\ndisplay(Math(latex('M^{(1)}= ')+latex(Me1.evalf(5))))\ndisplay(Latex('remeber that the x here is \"local\" to the element (i.e. from 0 to 100)'))\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "The bending moment in element 1:",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0x1374bd30>"
      },
      {
       "latex": "$$M^{(1)}= 950.0 x - 48333.0$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x1374bcd0>"
      },
      {
       "latex": "remeber that the x here is \"local\" to the element (i.e. from 0 to 100)",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0xa16d170>"
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Me2=M.subs({L:100, E:30e6, I:100, v1:D_n[2], phi1:D_n[3], v2:D_n[4], phi2:D_n[5] })\ndisplay(Latex('The bending moment in element 2:'))\ndisplay(Math(latex('M^{(2)}= ')+latex(Me2.evalf(5))))\ndisplay(Latex('remeber that the x here is \"local\" to the element (i.e. from 0 to 100)'))\n#remeber that the x here is \"local\" to the element (e.g. from 0 to 100)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "The bending moment in element 2:",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0x1374b5d0>"
      },
      {
       "latex": "$$M^{(2)}= - 1050.0 x + 53333.0$$",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Math at 0x1374b5d0>"
      },
      {
       "latex": "remeber that the x here is \"local\" to the element (i.e. from 0 to 100)",
       "metadata": {},
       "output_type": "display_data",
       "text": "<IPython.core.display.Latex at 0xa16d110>"
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    }
   ],
   "metadata": {}
  }
 ]
}