sympy_example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve for the nodal displacements and reaction force for the following spring element mesh:\n",
    "\n",
    "\n",
    "![](./images/fe_spring_mesh.png)\n",
    "\n",
    "take $k_1=k_2=k_3=k_4=k$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import the necessary libraries\n",
    "%matplotlib inline\n",
    "\n",
    "import sympy as sp  # import the sympy library\n",
    "sp.init_printing()  # printing equations using mathematical notation in the jupyter notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the variables that will be used \n",
    "delta1,delta2,delta3,delta4 = sp.symbols('delta_1,delta_2,delta_3,delta_4')\n",
    "R1,F = sp.symbols('R_1,F')\n",
    "k = sp.symbols('k')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}k & - k & 0 & 0\\\\- k & 3 k & - 2 k & 0\\\\0 & - 2 k & 3 k & - k\\\\0 & 0 & - k & k\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡k    -k    0    0 ⎤\n",
       "⎢                  ⎥\n",
       "⎢-k  3⋅k   -2⋅k  0 ⎥\n",
       "⎢                  ⎥\n",
       "⎢0   -2⋅k  3⋅k   -k⎥\n",
       "⎢                  ⎥\n",
       "⎣0    0     -k   k ⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define the stiffness matrix\n",
    "K = sp.Matrix([[k,-k,0,0],[-k,3*k,-2*k,0],[0,-2*k,3*k,-k],[0,0,-k,k]])\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\delta_{1}\\\\\\delta_{2}\\\\\\delta_{3}\\\\\\delta_{4}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡δ₁⎤\n",
       "⎢  ⎥\n",
       "⎢δ₂⎥\n",
       "⎢  ⎥\n",
       "⎢δ₃⎥\n",
       "⎢  ⎥\n",
       "⎣δ₄⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define the displacement vector\n",
    "u = sp.Matrix([delta1,delta2,delta3,delta4])\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}R_{1}\\\\0\\\\0\\\\F\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡R₁⎤\n",
       "⎢  ⎥\n",
       "⎢0 ⎥\n",
       "⎢  ⎥\n",
       "⎢0 ⎥\n",
       "⎢  ⎥\n",
       "⎣F ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define the force vector\n",
    "f = sp.Matrix([R1,0,0,F])\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ R_{1} : - F, \\quad \\delta_{2} : \\frac{F}{k}, \\quad \\delta_{3} : \\frac{3 F}{2 k}, \\quad \\delta_{4} : \\frac{5 F}{2 k}\\right \\}$$"
      ],
      "text/plain": [
       "⎧            F      3⋅F      5⋅F⎫\n",
       "⎨R₁: -F, δ₂: ─, δ₃: ───, δ₄: ───⎬\n",
       "⎩            k      2⋅k      2⋅k⎭"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Substitute delta1=0 into the system of equations and solve for the remaining unknowns\n",
    "sp.solve((K@u-f).subs(delta1,0),R1,delta2,delta3,delta4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
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   "language": "python",
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