winkler_foundation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ae4d301f-4524-41b6-a874-9c55def97c89",
   "metadata": {},
   "source": [
    "# Beam on Winkler Foundation Exact Solution and Finite Element Approximation\n",
    "\n",
    "## Requirements\n",
    "sympy  \n",
    "numpy  \n",
    "scipy  \n",
    "matplotlib  \n",
    "\n",
    "## Purpose\n",
    "This notebook does the following:\n",
    "\n",
    "1. Derive the 4th-order Ordinary Differential Equation (ODE) governing the response of a beam on elastic foundation\n",
    "2. Define the solution to the ODE\n",
    "3. Develop a solution strategy to solve for the coefficients to satsify boundary conditions\n",
    "4. Develop equations for the stiffness matrix of an Euler-Bernoulli beam using Hermite cubic polynomial shape functions\n",
    "5. Develop equations for the stiffness matrix of the p-y \"springs\" attached to the beam\n",
    "6. Approximate the solution using the finite element method"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72780871-5388-4649-ac06-bdff85bcb670",
   "metadata": {},
   "source": [
    "## Winkler Assumption\n",
    "\n",
    "The Winkler assumption is that a pressure develops when a beam displaces into an elastic medium, and that the pressure is proportional to the displacement of the beam, and independent of the stresses at other points on the beam. It's important to remember that the Winkler assumption is mechanically incorrect, but it's useful. If you analyze a beam using continuum methods, the pressure is not actually proportional to displacement. As an example, a rigid beam resting on an elastic halfspace has infinite pressure at the ends and smaller pressures in the middle, even though the displacements of the beam are identical everywhere. The solution is nevertheless useful, and forms the basis for laterally loaded pile analyses using programs like LPile.  \n",
    "\n",
    "## Definition of Terms  \n",
    "\n",
    "$EI$ = flexural stiffness of beam units = $\\left(FL^2\\right)$  \n",
    "$k_y^i$ = subgrade reaction stiffness intensity $\\left(F/L/L\\right)$  \n",
    "$\\beta = \\sqrt[^4]{\\frac{k_y^i}{4EI}}$ = Winkler stiffness parameter $\\left(L^{-1}\\right)$  \n",
    "$p$ = line load acting on beam  \n",
    "$V$ = shear in beam  \n",
    "$M$ = bending moment in beam  \n",
    "$\\phi$ = curvature of beam  \n",
    "$\\theta$ = rotation of beam  \n",
    "$u$ = beam displacement  \n",
    "$z$ = length dimension along beam axis  \n",
    "$H$ = length of beam\n",
    "<br>\n",
    "Note: The units for $k_y^i$ are often a source of confusion. I specify them as $\\left(F/L/L\\right)$ here because the stiffness intensity relates the line load mobilized against the beam to the beam displacement. So it's like a line load with units of $F/L$ will be mobilized for a displacement with units of $L$. However, the units are often presented as $\\left(F/L^2\\right)$.  \n",
    "\n",
    "## Governing Differential Equation  \n",
    "\n",
    "An Euler-Bernoulli beam is defined by the following relationships in integral form:  \n",
    "\n",
    "$V = \\int{p}$  \n",
    "$M = \\int{V} = EI\\phi$  \n",
    "$\\theta = \\int{\\phi}$  \n",
    "$u = \\int{\\theta}$  \n",
    "\n",
    "The equivalent expression in differential form are:  \n",
    "\n",
    "$\\theta = \\frac{du}{dz}$  \n",
    "$\\phi = \\frac{d^2u}{dz^2}$  \n",
    "$M = EI\\frac{d^2u}{dz^2}$  \n",
    "$V = EI\\frac{d^3u}{dz^3}$  \n",
    "$p = EI\\frac{d^4u}{dz^4}$  \n",
    "\n",
    "The Winkler assumption give us:\n",
    "\n",
    "$p = -k_y^iu$  \n",
    "\n",
    "Equating the expressions for p from beam theory with that for the Winkler assumption produces the governing differential equation:  \n",
    "\n",
    "$EI\\frac{d^4u}{dz^4} + k_y^iu = 0$  \n",
    "\n",
    "## Solution of Governing Differential Equation  \n",
    "\n",
    "Assuming that $EI$ and $k_y^i$ are constants, the analytical solution for displacement and derivatives (computed in the cell below using Sympy) takes the form:  \n",
    "\n",
    "$u = C_1e^{-\\beta z}cos\\left(\\beta z\\right) + C_2e^{-\\beta z}sin\\left(\\beta z\\right) + C_3e^{\\beta z}cos\\left(\\beta z\\right) + C_4e^{\\beta z}cos\\left(\\beta z\\right)$\n",
    "\n",
    "$\\frac{du}{dz} = - C_{1} \\beta e^{- \\beta x} \\sin{\\left(\\beta x \\right)} - C_{1} \\beta e^{- \\beta x} \\cos{\\left(\\beta x \\right)} - C_{2} \\beta e^{- \\beta x} \\sin{\\left(\\beta x \\right)} + C_{2} \\beta e^{- \\beta x} \\cos{\\left(\\beta x \\right)} - C_{3} \\beta e^{\\beta x} \\sin{\\left(\\beta x \\right)} + C_{3} \\beta e^{\\beta x} \\cos{\\left(\\beta x \\right)} + C_{4} \\beta e^{\\beta x} \\sin{\\left(\\beta x \\right)} + C_{4} \\beta e^{\\beta x} \\cos{\\left(\\beta x \\right)}$  \n",
    "\n",
    "$\\frac{d^2u}{dz^2} = 2 \\beta^{2} \\left(C_{1} e^{- \\beta x} \\sin{\\left(\\beta x \\right)} - C_{2} e^{- \\beta x} \\cos{\\left(\\beta x \\right)} - C_{3} e^{\\beta x} \\sin{\\left(\\beta x \\right)} + C_{4} e^{\\beta x} \\cos{\\left(\\beta x \\right)}\\right)$  \n",
    "\n",
    "$\\frac{d^3u}{dz^3} = 2 \\beta^{3} \\left(- C_{1} e^{- \\beta x} \\sin{\\left(\\beta x \\right)} + C_{1} e^{- \\beta x} \\cos{\\left(\\beta x \\right)} + C_{2} e^{- \\beta x} \\sin{\\left(\\beta x \\right)} + C_{2} e^{- \\beta x} \\cos{\\left(\\beta x \\right)} - C_{3} e^{\\beta x} \\sin{\\left(\\beta x \\right)} - C_{3} e^{\\beta x} \\cos{\\left(\\beta x \\right)} - C_{4} e^{\\beta x} \\sin{\\left(\\beta x \\right)} + C_{4} e^{\\beta x} \\cos{\\left(\\beta x \\right)}\\right)$  \n",
    "\n",
    "$\\frac{d^4u}{dz^4} = - 4 \\beta^{4} \\left(C_{1} e^{- \\beta x} \\cos{\\left(\\beta x \\right)} + C_{2} e^{- \\beta x} \\sin{\\left(\\beta x \\right)} + C_{3} e^{\\beta x} \\cos{\\left(\\beta x \\right)} + C_{4} e^{\\beta x} \\sin{\\left(\\beta x \\right)}\\right)$  \n",
    "\n",
    "The C's are coefficients that must be solved to satisfy essential boundary conditions, generally specified at the ends of the beam (i.e., at $z=0$ and at $z=L$).  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "10f2dedc-57a3-4a2d-a594-9b64f5b90a3e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- 4 \\beta^{4} \\left(C_{1} e^{- \\beta x} \\cos{\\left(\\beta x \\right)} + C_{2} e^{- \\beta x} \\sin{\\left(\\beta x \\right)} + C_{3} e^{\\beta x} \\cos{\\left(\\beta x \\right)} + C_{4} e^{\\beta x} \\sin{\\left(\\beta x \\right)}\\right)\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 4 \\beta^{4} \\left(C_{1} e^{- \\beta x} \\cos{\\left(\\beta x \\right)} + C_{2} e^{- \\beta x} \\sin{\\left(\\beta x \\right)} + C_{3} e^{\\beta x} \\cos{\\left(\\beta x \\right)} + C_{4} e^{\\beta x} \\sin{\\left(\\beta x \\right)}\\right)$"
      ],
      "text/plain": [
       "    4 ⎛    -β⋅x                -β⋅x                β⋅x                β⋅x     \n",
       "-4⋅β ⋅⎝C₁⋅ℯ    ⋅cos(β⋅x) + C₂⋅ℯ    ⋅sin(β⋅x) + C₃⋅ℯ   ⋅cos(β⋅x) + C₄⋅ℯ   ⋅sin(\n",
       "\n",
       "    ⎞\n",
       "β⋅x)⎠"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "\n",
    "x = sym.Symbol('x')\n",
    "beta = sym.Symbol('beta')\n",
    "C1 = sym.Symbol('C1')\n",
    "C2 = sym.Symbol('C2')\n",
    "C3 = sym.Symbol('C3')\n",
    "C4 = sym.Symbol('C4')\n",
    "sym.init_printing()\n",
    "f = C1*sym.exp(-beta*x)*sym.cos(beta*x) + C2*sym.exp(-beta*x)*sym.sin(beta*x) + C3*sym.exp(beta*x)*sym.cos(beta*x) + C4*sym.exp(beta*x)*sym.sin(beta*x)\n",
    "X = sym.diff(f,x,4)\n",
    "sym.print_latex(X)\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3b79441e-6659-40bb-bcdd-b9d9ac6e954f",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Function to return displacement or its derivatives, up to the 4th derivative\n",
    "def get_u(x,beta,C,order):\n",
    "    bx = beta*x\n",
    "    C1 = C[0]\n",
    "    C2 = C[1]\n",
    "    C3 = C[2]\n",
    "    C4 = C[3]\n",
    "    if(order==0):\n",
    "        output = C1*np.exp(-bx)*np.cos(bx)\n",
    "        output += C2*np.exp(-bx)*np.sin(bx)\n",
    "        output += C3*np.exp(bx)*np.cos(bx)\n",
    "        output += C4*np.exp(bx)*np.sin(bx)\n",
    "    elif(order==1):\n",
    "        output = C1*beta*(-np.exp(-bx)*np.sin(bx)-np.exp(-bx)*np.cos(bx))\n",
    "        output += C2*beta*(-np.exp(-bx)*np.sin(bx)+np.exp(-bx)*np.cos(bx))\n",
    "        output += C3*beta*(-np.exp(bx)*np.sin(bx)+np.exp(bx)*np.cos(bx))\n",
    "        output += C4*beta*(np.exp(bx)*np.sin(bx)+np.exp(bx)*np.cos(bx))\n",
    "    elif(order==2):\n",
    "        output = 2*beta**2*C1*np.exp(-bx)*np.sin(bx)\n",
    "        output += -2*beta**2*C2*np.exp(-bx)*np.cos(bx)\n",
    "        output += -2*beta**2*C3*np.exp(bx)*np.sin(bx)\n",
    "        output += 2*beta**2*C4*np.exp(bx)*np.cos(bx)\n",
    "    elif(order==3):\n",
    "        output = 2*C1*beta**3*(-np.exp(-bx)*np.sin(bx)+np.exp(-bx)*np.cos(bx))\n",
    "        output += 2*C2*beta**3*(np.exp(-bx)*np.sin(bx)+np.exp(-bx)*np.cos(bx))\n",
    "        output += 2*C3*beta**3*(-np.exp(bx)*np.sin(bx)-np.exp(bx)*np.cos(bx))\n",
    "        output += 2*C4*beta**3*(-np.exp(bx)*np.sin(bx)+np.exp(bx)*np.cos(bx))\n",
    "    elif(order==4):\n",
    "        output = -4*beta**4*C1*np.exp(-bx)*np.cos(bx)\n",
    "        output += -4*beta**4*C2*np.exp(-bx)*np.sin(bx)\n",
    "        output += -4*beta**4*C3*np.exp(bx)*np.cos(bx)\n",
    "        output += -4*beta**4*C4*np.exp(bx)*np.sin(bx)\n",
    "    else:\n",
    "        output = 'Invalid order in u(x,beta,C,order)'\n",
    "    return output\n",
    "\n",
    "# Function to solve for C's by setting up a matrix and vector based on specified boundary conditions\n",
    "def get_C(BCs,beta):\n",
    "    Mat = np.zeros((4,4))\n",
    "    Vec = np.zeros(4)\n",
    "    for i in range(4):\n",
    "        x = BCs[i][2]\n",
    "        bx = beta*x\n",
    "        if(BCs[i][0] == 0):\n",
    "            Mat[i][0] = np.exp(-bx)*np.cos(bx)\n",
    "            Mat[i][1] = np.exp(-bx)*np.sin(bx)\n",
    "            Mat[i][2] = np.exp(bx)*np.cos(bx)\n",
    "            Mat[i][3] = np.exp(bx)*np.sin(bx)\n",
    "        elif(BCs[i][0] == 1):\n",
    "            Mat[i][0] = beta*(-np.exp(-bx)*np.sin(bx)-np.exp(-bx)*np.cos(bx))\n",
    "            Mat[i][1] = beta*(-np.exp(-bx)*np.sin(bx)+np.exp(-bx)*np.cos(bx))\n",
    "            Mat[i][2] = beta*(-np.exp(bx)*np.sin(bx)+np.exp(bx)*np.cos(bx))\n",
    "            Mat[i][3] = beta*(np.exp(bx)*np.sin(bx)+np.exp(bx)*np.cos(bx))\n",
    "        elif(BCs[i][0] == 2):\n",
    "            Mat[i][0] = 2*beta**2*np.exp(-bx)*np.sin(bx)\n",
    "            Mat[i][1] = -2*beta**2*np.exp(-bx)*np.cos(bx)\n",
    "            Mat[i][2] = -2*beta**2*np.exp(bx)*np.sin(bx)\n",
    "            Mat[i][3] = 2*beta**2*np.exp(bx)*np.cos(bx)\n",
    "        elif(BCs[i][0] == 3):\n",
    "            Mat[i][0] = 2*beta**3*(-np.exp(-bx)*np.sin(bx)+np.exp(-bx)*np.cos(bx))\n",
    "            Mat[i][1] = 2*beta**3*(np.exp(-bx)*np.sin(bx)+np.exp(-bx)*np.cos(bx))\n",
    "            Mat[i][2] = 2*beta**3*(-np.exp(bx)*np.sin(bx)-np.exp(bx)*np.cos(bx))\n",
    "            Mat[i][3] = 2*beta**3*(-np.exp(bx)*np.sin(bx)+np.exp(bx)*np.cos(bx))\n",
    "        else:\n",
    "            return('Invalid order in get_C(BCs,beta)')\n",
    "        Vec[i] = BCs[i][1]\n",
    "    return np.linalg.solve(Mat,Vec)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a78b17b5-57d4-4b05-ad0e-9ccec3c100cc",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 5 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "'''Boundary conditions. Specify numpy array with four entries; one for each BC. \n",
    "First element indicates BC type (0 = disp, 1 = slope, 2 = moment, 3 = shear)\n",
    "Second element indicates BC value\n",
    "Third element indicates BC position\n",
    "e.g. [0, 0.1, 0.0] specifies displacement = 0 at z=0.\n",
    "     [1, 0.0, H] specifies rotation = 0 at z=H.\n",
    "'''\n",
    "\n",
    "H = 4\n",
    "EI = 1000\n",
    "kyi = 100\n",
    "beta = (kyi/(4*EI))**0.25\n",
    "BCs = np.asarray([\n",
    "    [0,0.1,0.0],\n",
    "    [1,0.0,0.0],\n",
    "    [0,0.0,H],\n",
    "    [1,0.0,H]    \n",
    "])\n",
    "C = get_C(BCs,beta)\n",
    "x = np.linspace(0,H,100)\n",
    "u = get_u(x,beta,C,0)\n",
    "theta = get_u(x,beta,C,1)\n",
    "M = EI*get_u(x,beta,C,2)\n",
    "V = EI*get_u(x,beta,C,3)\n",
    "p = EI*get_u(x,beta,C,4)\n",
    "fig, ax = plt.subplots(ncols=5,sharey='row',figsize=(8,4))\n",
    "ax[0].plot(u,x)\n",
    "ax[1].plot(theta,x)\n",
    "ax[2].plot(M,x)\n",
    "ax[3].plot(V,x)\n",
    "ax[4].plot(p,x,label='Beam')\n",
    "ax[4].plot(-kyi*u,x,label='Winkler')\n",
    "ax[4].legend()\n",
    "ax[0].invert_yaxis()\n",
    "ax[0].set_ylabel('depth')\n",
    "ax[0].set_xlabel('u')\n",
    "ax[1].set_xlabel(r'$\\theta$')\n",
    "ax[2].set_xlabel('M')\n",
    "ax[3].set_xlabel('V')\n",
    "ax[4].set_xlabel('p')\n",
    "for a in ax:\n",
    "    a.grid(True, alpha=0.5)\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85853f2c-dec5-4a05-a742-eb18194bf19e",
   "metadata": {},
   "source": [
    "# Finite Element Approximate Solution  \n",
    "\n",
    "The finite element approximation requires definition of a stiffness matrix for the basic element, which consists of an Euler-Bernoulli beam supported by two Winkler springs, as illustrated below. Nodes 1 and 2 have two degrees of freedom; one for horizontal displacement and one for rotation. Nodes 3 and 4 are assumed to be fixed (i.e., no free-field ground displacement).\n",
    "\n",
    "<img src='https://raw.githubusercontent.com/sjbrandenberg/cee227/main/WinklerElement.png' alt='Winkler Element' style='height:200px;'>\n",
    "\n",
    "The beam stiffness matrix is developed using Hermite cubic polynomial shape functions. These shape functions do not match the analytical solution for constant $k_y^i$ and $EI$, but are reasonably accurate when an adequate number of elements is utilized, and/or when the wall is stiff relative to the soil."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "1c9966ad-a837-47fe-828c-7b68eece997c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "dx = 1\n",
    "N = 100\n",
    "z = np.linspace(0,dx,N)\n",
    "phi = np.zeros((4,N))\n",
    "phi[0] = (1-z/dx)**2*(1+2*z/dx)\n",
    "phi[1] = z*(1-z/dx)**2\n",
    "phi[2] = (z/dx)**2*(3-2*z/dx)\n",
    "phi[3] = -z**2/dx*(1-z/dx)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "for i in range(4):\n",
    "    ax.plot(phi[i],z,label=str(i+1))\n",
    "ax.legend(title='Shape Function')\n",
    "ax.set_ylabel(r'$\\phi$')\n",
    "ax.set_xlabel('z/dx')\n",
    "ax.grid(True,alpha=0.5)\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b43bcee-e69f-4f61-8c61-95c25b18be02",
   "metadata": {},
   "source": [
    "## Beam Stiffness Matrix\n",
    "\n",
    "The beam stiffness matrix is computed as $K_{ij} = \\int_0^H EI\\frac{d^2\\phi_i}{dz^2}\\frac{d^2\\phi_j}{dz^2}dz$. Solving derivatives using Sympy in the cell below and factoring out $EI/H^3$, we obtain the following:  \n",
    "\n",
    "<br>\n",
    "\n",
    "$\\begin{Bmatrix}\n",
    "F_1\\\\\n",
    "M_1\\\\\n",
    "F_2\\\\\n",
    "M_2\n",
    "\\end{Bmatrix}$ = $\\frac{EI}{H^3}$\n",
    "$\\begin{bmatrix}\n",
    "12 & 6H & -12 & 6H\\\\\n",
    "6H & 4H^2 & -6H & 2H^2\\\\\n",
    "-12 & -6H & 12 & -6H\\\\\n",
    "6L & 2H^2 & -6H & 4H^2\n",
    "\\end{bmatrix}$\n",
    "$\\begin{Bmatrix}\n",
    "u_1\\\\\n",
    "\\theta_1\\\\\n",
    "u_2\\\\\n",
    "\\theta_2\n",
    "\\end{Bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2c276553-1e3e-4c1b-bc36-0b63d4362834",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{12 EI}{H^{3}} & \\frac{6 EI}{H^{2}} & - \\frac{12 EI}{H^{3}} & \\frac{6 EI}{H^{2}}\\\\\\frac{6 EI}{H^{2}} & \\frac{4 EI}{H} & - \\frac{6 EI}{H^{2}} & \\frac{2 EI}{H}\\\\- \\frac{12 EI}{H^{3}} & - \\frac{6 EI}{H^{2}} & \\frac{12 EI}{H^{3}} & - \\frac{6 EI}{H^{2}}\\\\\\frac{6 EI}{H^{2}} & \\frac{2 EI}{H} & - \\frac{6 EI^{2}}{H^{2}} & \\frac{4 EI}{H}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 12⋅EI    6⋅EI   -12⋅EI    6⋅EI ⎤\n",
       "⎢ ─────    ────   ───────   ──── ⎥\n",
       "⎢    3       2        3       2  ⎥\n",
       "⎢   H       H        H       H   ⎥\n",
       "⎢                                ⎥\n",
       "⎢ 6⋅EI     4⋅EI   -6⋅EI     2⋅EI ⎥\n",
       "⎢ ────     ────   ──────    ──── ⎥\n",
       "⎢   2       H        2       H   ⎥\n",
       "⎢  H                H            ⎥\n",
       "⎢                                ⎥\n",
       "⎢-12⋅EI   -6⋅EI    12⋅EI   -6⋅EI ⎥\n",
       "⎢───────  ──────   ─────   ──────⎥\n",
       "⎢    3       2        3       2  ⎥\n",
       "⎢   H       H        H       H   ⎥\n",
       "⎢                                ⎥\n",
       "⎢                      2         ⎥\n",
       "⎢ 6⋅EI     2⋅EI   -6⋅EI     4⋅EI ⎥\n",
       "⎢ ────     ────   ───────   ──── ⎥\n",
       "⎢   2       H         2      H   ⎥\n",
       "⎣  H                 H           ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = sym.Symbol('z')\n",
    "H = sym.Symbol('H')\n",
    "EI = sym.Symbol('EI')\n",
    "phi1 = (1-z/H)*(1-z/H)*(1+2*z/H)\n",
    "phi2 = z*(1-z/H)*(1-z/H)\n",
    "phi3 = (z/H)*(z/H)*(3-2*z/H)\n",
    "phi4 = -z*z/H*(1-z/H)\n",
    "\n",
    "K11 = EI*sym.integrate(sym.diff(phi1,z,2)*sym.diff(phi1,z,2),(z,0,H))\n",
    "K12 = EI*sym.integrate(sym.diff(phi1,z,2)*sym.diff(phi2,z,2),(z,0,H))\n",
    "K13 = EI*sym.integrate(sym.diff(phi1,z,2)*sym.diff(phi3,z,2),(z,0,H))\n",
    "K14 = EI*sym.integrate(sym.diff(phi1,z,2)*sym.diff(phi4,z,2),(z,0,H))\n",
    "\n",
    "K21 = EI*sym.integrate(sym.diff(phi2,z,2)*sym.diff(phi1,z,2),(z,0,H))\n",
    "K22 = EI*sym.integrate(sym.diff(phi2,z,2)*sym.diff(phi2,z,2),(z,0,H))\n",
    "K23 = EI*sym.integrate(sym.diff(phi2,z,2)*sym.diff(phi3,z,2),(z,0,H))\n",
    "K24 = EI*sym.integrate(sym.diff(phi2,z,2)*sym.diff(phi4,z,2),(z,0,H))\n",
    "\n",
    "K31 = EI*sym.integrate(sym.diff(phi3,z,2)*sym.diff(phi1,z,2),(z,0,H))\n",
    "K32 = EI*sym.integrate(sym.diff(phi3,z,2)*sym.diff(phi2,z,2),(z,0,H))\n",
    "K33 = EI*sym.integrate(sym.diff(phi3,z,2)*sym.diff(phi3,z,2),(z,0,H))\n",
    "K34 = EI*sym.integrate(sym.diff(phi3,z,2)*sym.diff(phi4,z,2),(z,0,H))\n",
    "\n",
    "K41 = EI*sym.integrate(sym.diff(phi4,z,2)*sym.diff(phi1,z,2),(z,0,H))\n",
    "K42 = EI*sym.integrate(sym.diff(phi4,z,2)*sym.diff(phi2,z,2),(z,0,H))\n",
    "K43 = EI*EI*sym.integrate(sym.diff(phi4,z,2)*sym.diff(phi3,z,2),(z,0,H))\n",
    "K44 = EI*sym.integrate(sym.diff(phi4,z,2)*sym.diff(phi4,z,2),(z,0,H))\n",
    "\n",
    "sym.Matrix([[K11,K12,K13,K14],[K21,K22,K23,K24],[K31,K32,K33,K34],[K41,K42,K43,K44]])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e33e11e-857e-430e-90ba-137fe0964277",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Winkler Stiffness Matrix  \n",
    "\n",
    "The stiffness intensity $k_y^i$ has units of $F/L/L$, and the intensity must be divided between the two p-y elements. Using the tributary length method, the stiffness of each p-y element is $k_y^i dz/2$. Noting that forces are positive from left-to-right, we obtain the following stiffness matrix:  \n",
    "\n",
    "$\\begin{Bmatrix}\n",
    "F_1\\\\\n",
    "F_2\\\\\n",
    "F_3\\\\\n",
    "F_4\n",
    "\\end{Bmatrix}$ = $\\frac{k_y^i dz}{2}$\n",
    "$\\begin{bmatrix}\n",
    "1 & 0 & -1 & 0\\\\\n",
    "0 & 1 & 0 & -1\\\\\n",
    "-1 & 0 & 1 & 0\\\\\n",
    "0 & -1 & 0 & 1\n",
    "\\end{bmatrix}$\n",
    "$\\begin{Bmatrix}\n",
    "u_1\\\\\n",
    "u_2\\\\\n",
    "u_3\\\\\n",
    "u_4\n",
    "\\end{Bmatrix}$  \n",
    "\n",
    "Noting that $u_3$ = $u_4$ = 0, we can condense out the bottom two rows and right two columns. The resulting equation is:\n",
    "\n",
    "$\\begin{Bmatrix}\n",
    "F_1\\\\\n",
    "F_2\n",
    "\\end{Bmatrix}$ = $\\frac{k_y^i dz}{2}$\n",
    "$\\begin{bmatrix}\n",
    "1 & 0\\\\\n",
    "0 & 1\\\\\n",
    "\\end{bmatrix}$\n",
    "$\\begin{Bmatrix}\n",
    "u_1\\\\\n",
    "u_2\n",
    "\\end{Bmatrix}$  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3b26a88-496a-4e6a-9c15-48ab31f6188a",
   "metadata": {},
   "source": [
    "## Combined Stiffness Matrix\n",
    "\n",
    "We can now combine the two stiffness matrices into a single matrix by combining contributions to nodes 1 and 2:  \n",
    "\n",
    "$\\begin{Bmatrix}\n",
    "F_1\\\\\n",
    "M_1\\\\\n",
    "F_2\\\\\n",
    "M_2\n",
    "\\end{Bmatrix}$ = \n",
    "$\\begin{bmatrix}\n",
    "12\\frac{EI}{H^3} + \\frac{k_y^i dz}{2} & 6\\frac{EI}{H^2} & -12\\frac{EI}{H^3} & 6\\frac{EI}{H^2}\\\\\n",
    "6\\frac{EI}{H^2} & 4\\frac{EI}{H} & -6\\frac{EI}{H^2} & 2\\frac{EI}{H} \\\\\n",
    "-12\\frac{EI}{H^3} & -6\\frac{EI}{H^2} & 12\\frac{EI}{H^3} + \\frac{k_y^i dz}{2} & -6\\frac{EI}{H^2}\\\\\n",
    "6\\frac{EI}{H^2} & 2\\frac{EI}{H} & -6\\frac{EI}{H^2} & 4\\frac{EI}{H}\n",
    "\\end{bmatrix}$\n",
    "$\\begin{Bmatrix}\n",
    "u_1\\\\\n",
    "\\theta_1\\\\\n",
    "u_2\\\\\n",
    "\\theta_2\n",
    "\\end{Bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "52b72aff-06f1-456a-a06d-2ccfdc6a5eac",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Element stiffness matrix equations\n",
    "# Individual equations are provided for the beam-column matrix and for the p-y matrix because they are useful later for computing internal forces\n",
    "def get_Kbeam(EI,H):\n",
    "    Kbeam = EI/H**3*np.asarray([\n",
    "        [12, 6*H, -12, 6*H],\n",
    "        [6*H, 4*H**2, -6*H, 2*H**2],\n",
    "        [-12, -6*H, 12, -6*H],\n",
    "        [6*H, 2*H**2, -6*H, 4*H**2]        \n",
    "    ])\n",
    "    return Kbeam\n",
    "\n",
    "def get_Kpy(kyi,H):\n",
    "    Kpy = kyi*H/2.0*np.asarray([\n",
    "        [1.0, 0.0, 0.0, 0.0],\n",
    "        [0.0, 0.0, 0.0, 0.0],\n",
    "        [0.0, 0.0, 1.0, 0.0],\n",
    "        [0.0, 0.0, 0.0, 0.0]\n",
    "    ])\n",
    "    return Kpy\n",
    "    \n",
    "def get_Kele(EI,kpy,H):\n",
    "    Kele = get_Kbeam(EI,H) + get_Kpy(kyi,H)\n",
    "    return Kele"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cb9b392-f37a-4c7c-92db-749bd7da83f2",
   "metadata": {},
   "source": [
    "## Assembling global stiffness matrix from element stiffness matrices\n",
    "\n",
    "After forming the stiffness matrix for each element, they must be combined into a global stiffness matrix based on nodal connectivity. For simplicity we herein assume that the nodes are numbered in order such that the beam column connects 1 to 2, 2 to 3, and so forth, and also that $dz$, $EI$, and $k_y^i$ are constant. We later provide an example with variable $EI$ and $k_y^i$ values. The approach adopted here is to initialize the global stiffness matrix $K_{global}$ to zeros, and add the element stiffness matrices to it in a loop over all elements."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "6a8c1c9d-06d5-4e64-842b-385ec3f7dfa1",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Now solve using finite element method\n",
    "\n",
    "# Input parameters\n",
    "Nele = 10                         # Number of beam elements\n",
    "H = 4.0                           # Total length of beam\n",
    "EI = 1000                         # Flexural stiffness of beam\n",
    "kyi = 100                         # Winkler stiffness intensity\n",
    "dx2 = H/Nele                      # Element length\n",
    "x2 = np.linspace(0,H,Nele+1)      # Array of nodal locations\n",
    "x2_ele = np.zeros(2*Nele)\n",
    "x2_ele[0::2] = x2[0:Nele]\n",
    "x2_ele[1::2] = x2[1:Nele+1]\n",
    "ele_sign = np.zeros(2*Nele)\n",
    "ele_sign[0::2] = 1\n",
    "ele_sign[1::2] = -1\n",
    "# Boundary conditions array. This array is a bit different than for the analytical solution.\n",
    "'''Boundary conditions. Specify numpy array with four entries; one for each BC. \n",
    "First element indicates BC type (0 = disp, 1 = slope, 2 = moment, 3 = shear)\n",
    "Second element indicates BC value\n",
    "Third element indicates node number\n",
    "e.g. [0, 0.1, 0.0] specifies displacement = 0 at z=0.\n",
    "     [1, 0.0, H] specifies rotation = 0 at z=L.\n",
    "'''\n",
    "\n",
    "Kglobal = np.zeros((2*(Nele+1),2*(Nele+1)))  \n",
    "I = 0\n",
    "for k in range(Nele):\n",
    "    Kele = get_Kele(EI,kyi,dx2)\n",
    "    for i in range(4):\n",
    "        for j in range(4):\n",
    "            Kglobal[i+I][j+I] += Kele[i][j]\n",
    "    I+=2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "676a699b-ef6d-4f9a-8b32-4d9bf769997e",
   "metadata": {},
   "source": [
    "## Solving the System of Equations\n",
    "\n",
    "Before solving the system of equations, a force array is constructed from the imposed force boundary conditions, and a displacement array is constructed from the imposed displacement boundary conditions. The displacement boundary conditions are then condensed out of the force vector and global stiffness matrix to solve for the unknown displacements. The unknown displacements are then inserted into the displacement array. Finally, internal beam forces are computed using the individual beam element stiffness matrices, and the subgrade reaction pressures are computed from the Winkler spring forces divided by the tributary length."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "eb2796dd-9ca0-4bb3-990e-fe9921c3787c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 5 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Define boundary conditions\n",
    "BCs = np.asarray([\n",
    "    [0,0.1,1],\n",
    "    [1,0.0,1],\n",
    "    [0,0.0,Nele+1],\n",
    "    [1,0.0,Nele+1]    \n",
    "])\n",
    "\n",
    "BCs = np.asarray([\n",
    "    [3,33.2510024267067,1],\n",
    "    [2,45.4483024248866,1],\n",
    "    [0,0.0,Nele+1],\n",
    "    [1,0.0,Nele+1]    \n",
    "])\n",
    "\n",
    "# Create force boundary condition array\n",
    "F = np.zeros(2*(Nele+1))\n",
    "for bc in BCs:\n",
    "    if(bc[0] == 3):\n",
    "        F[int(2*(bc[2]-1))] = bc[1]\n",
    "    elif(bc[0] == 2):\n",
    "        F[int(2*(bc[2]-1)+1)] = bc[1]\n",
    "\n",
    "# Create displacement boundary condition array\n",
    "u2 = np.zeros(2*(Nele+1))\n",
    "for bc in BCs:\n",
    "    if(bc[0] == 0):\n",
    "        u2[int(2*(bc[2]-1))] = bc[1]\n",
    "    elif(bc[0] == 1):\n",
    "        u2[int(2*(bc[2]-1)+1)] = bc[1]\n",
    "\n",
    "# Now impose essential BC's by condensing matrix, and solve for unknown displacements\n",
    "F = F - Kglobal@u2  # compute forces caused by essential boundary conditions\n",
    "mask = np.full(2*(Nele+1),True)\n",
    "for bc in BCs:\n",
    "    if(bc[0]==0):\n",
    "        mask[int(2*(bc[2]-1))] = False\n",
    "    elif(bc[0]==1):\n",
    "        mask[int(2*(bc[2]-1)+1)] = False\n",
    "u_new = np.linalg.solve(Kglobal[mask].T[mask].T,F[mask])\n",
    "\n",
    "# assemble the new displacements into the overall displacement array\n",
    "u2[mask] = u_new\n",
    "\n",
    "# assemble internal forces based on individual element stiffness matrices\n",
    "F_beam = np.zeros(4*Nele)\n",
    "F_py = np.zeros(4*Nele)\n",
    "I = 0\n",
    "for i in range(Nele):\n",
    "    Kbeam = get_Kbeam(EI,dx2)\n",
    "    F_beam[2*I:2*I+4] = Kbeam@u2[I:I+4]\n",
    "    Kpy = get_Kpy(kyi,dx2)\n",
    "    F_py[2*I:2*I+4] = Kpy@u2[I:I+4]\n",
    "    I+=2\n",
    "fig, ax = plt.subplots(ncols=5,sharey='row',figsize=(8,4))\n",
    "ax[0].plot(u,x,label='exact')\n",
    "ax[0].plot(u2[0::2],x2,label='FEM')\n",
    "ax[1].plot(theta,x)\n",
    "ax[1].plot(u2[1::2],x2)\n",
    "ax[2].plot(M,x)\n",
    "ax[2].plot(F_beam[1::2]*-ele_sign,x2_ele)\n",
    "ax[3].plot(V,x)\n",
    "ax[3].plot(F_beam[0::2]*ele_sign,x2_ele)\n",
    "ax[4].plot(p,x)\n",
    "ax[4].plot(-2*F_py[0::2]/dx2,x2_ele)\n",
    "ax[0].legend()\n",
    "ax[0].invert_yaxis()\n",
    "ax[0].set_ylabel('depth')\n",
    "ax[0].set_xlabel('u')\n",
    "ax[1].set_xlabel(r'$\\theta$')\n",
    "ax[2].set_xlabel('M')\n",
    "ax[3].set_xlabel('V')\n",
    "ax[4].set_xlabel('p')\n",
    "for a in ax:\n",
    "    a.grid(True, alpha=0.5)\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebabbc80-99c6-420a-b90b-11accc47cc2d",
   "metadata": {
    "tags": []
   },
   "source": [
    "## A few comments\n",
    "\n",
    "You might notice that the shear distribution is piece-wise constant, with discontinuities at the node locations. The reason for this behavior is that the Winkler stiffness intensity is lumped at the nodes, and no forces act on the beam elements in between the nodes. This is an outcome of using Hermite cubic polynomial shape functions, which have zero pressures acting on the beam between the nodes. Furthermore, the shear at the top of the beam is less than the exact solution. That's because when a displacement is imposed on top of the beam, some of the force goes into the beam element and some goes into the Winkler spring. The orange line above represents the shear in the beam. These differences become smaller when more elements are utilized."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09e37278-a229-4358-b14c-78a88d7c78f3",
   "metadata": {},
   "source": [
    "## Solution with vertically variable $k_y^i$\n",
    "\n",
    "The solution above has an analytical solution, which is nice because we can validate the finite element implementation against a known solution. However, we would never need the finite element method if every problem was so simple. The script below develops a solution for a vertically variable Winkler stiffness intensity. We adopt the form utilized by [Brandenberg et al (2020)](https://ascelibrary.org/doi/10.1061/%28ASCE%29GT.1943-5606.0002374):  \n",
    "\n",
    "$k_y^i\\left(z\\right) = k_{yH}^i\\left[b+\\left(1+b\\right)\\frac{z}{H}\\right]^{2n}$  \n",
    "\n",
    "where $k_{yH}^i$ = Winkler stiffness intensity at the bottom of the beam, $b$ and $n$ are model coefficients that control the shape of the variation of $k_y^i$ with depth, and $H$ is the length of the beam. For static conditions, a rigid wall, and an infinitely long soil deposit, the value of $k_{yH}^i$ is given by:  \n",
    "\n",
    "$k_{yH}^i = \\frac{G_H}{H}\\frac{2}{\\sqrt{\\left(1-\\nu\\right)\\left(2-\\nu\\right)}}\\left(1.06 e^{-1.97\\left(1-2n\\right)-3.01b}+\\frac{\\pi}{2}\\right)$  \n",
    "\n",
    "We adopt the following parameters:\n",
    "\n",
    "$G_H = \\rho V_S^2$  \n",
    "$\\rho$ = 2.0 $Mg/m^3$  \n",
    "$V_{SH}$ = 200 m/s  \n",
    "$b$ = 0.1  \n",
    "$n$ = 0.25  \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "0fbffe05-72ae-4b2b-9972-29b142bcb768",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def get_kyi(kyiH,b,z,H,n):\n",
    "    return kyiH*(b+(1+b)*z/H)**(2*n)\n",
    "\n",
    "rho = 2.0\n",
    "VsH = 200\n",
    "GH = rho*VsH**2\n",
    "b = 0.1\n",
    "n = 0.25\n",
    "nu = 0.3\n",
    "kyiH = GH/H*2/np.sqrt((1-nu)*(2-nu))*(1.06*np.exp(-1.97*(1-2*n)-3.01*b)+np.pi/2)\n",
    "Vs = VsH*(b+(1-b)*x/H)**n\n",
    "kyi = get_kyi(kyiH,b,x,H,n)\n",
    "\n",
    "fig, ax = plt.subplots(ncols=2, sharey='row')\n",
    "ax[0].plot(Vs,x)\n",
    "ax[1].plot(kyi,x)\n",
    "ax[0].invert_yaxis()\n",
    "ax[0].set_ylabel(r'$z$ (m)')\n",
    "ax[0].set_xlabel(r'$V_S$ (m/s)')\n",
    "ax[1].set_xlabel(r'$k_y^i$ (kN/m/m)')\n",
    "for a in ax:\n",
    "    a.grid(True,alpha=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "58163dda-48a4-4273-95a3-b8b4b94b30d7",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 5 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Nele = 100                        # Number of beam elements\n",
    "H = 4.0                           # Total length of beam\n",
    "EI = 100000                       # Flexural stiffness of beam\n",
    "dx2 = H/Nele                      # Element length\n",
    "x2 = np.linspace(0,H,Nele+1)      # Array of nodal locations\n",
    "x2_ele = np.zeros(2*Nele)\n",
    "x2_ele[0::2] = x2[0:Nele]\n",
    "x2_ele[1::2] = x2[1:Nele+1]\n",
    "ele_sign = np.zeros(2*Nele)\n",
    "ele_sign[0::2] = 1\n",
    "ele_sign[1::2] = -1\n",
    "\n",
    "# We now have to define the stiffness matrix for variable kyi\n",
    "def get_Kpy_variable(kyiH,b,z1,z2,H,n):\n",
    "    kyi1 = get_kyi(kyiH,b,z1,H,n)\n",
    "    kyi2 = get_kyi(kyiH,b,z2,H,n)\n",
    "    dz = np.abs(z2-z1)\n",
    "    Kpy = dz/2.0*np.asarray([\n",
    "        [kyi1, 0.0, 0.0, 0.0],\n",
    "        [0.0, 0.0, 0.0, 0.0],\n",
    "        [0.0, 0.0, kyi2, 0.0],\n",
    "        [0.0, 0.0, 0.0, 0.0]\n",
    "    ])\n",
    "    return Kpy\n",
    "\n",
    "# Define global force vector\n",
    "Kglobal = np.zeros((2*(Nele+1),2*(Nele+1)))  \n",
    "I = 0\n",
    "for k in range(Nele):\n",
    "    Kbeam = get_Kbeam(EI,dx2)\n",
    "    Kpy = get_Kpy_variable(kyiH,b,x2_ele[2*k],x2_ele[2*k+1],H,n)\n",
    "    Kele = Kbeam + Kpy\n",
    "    for i in range(4):\n",
    "        for j in range(4):\n",
    "            Kglobal[i+I][j+I] += Kele[i][j]\n",
    "    I+=2\n",
    "\n",
    "# Define boundary conditions\n",
    "BCs = np.asarray([\n",
    "    [0,0.1,1],\n",
    "    [1,0.0,1],\n",
    "    [0,0.0,Nele+1],\n",
    "    [1,0.0,Nele+1]    \n",
    "])\n",
    "\n",
    "# Create force boundary condition array\n",
    "F = np.zeros(2*(Nele+1))\n",
    "for bc in BCs:\n",
    "    if(bc[0] == 3):\n",
    "        F[int(2*(bc[2]-1))] = bc[1]\n",
    "    elif(bc[0] == 2):\n",
    "        F[int(2*(bc[2]-1)+1)] = bc[1]\n",
    "\n",
    "# Create displacement boundary condition array\n",
    "u2 = np.zeros(2*(Nele+1))\n",
    "for bc in BCs:\n",
    "    if(bc[0] == 0):\n",
    "        u2[int(2*(bc[2]-1))] = bc[1]\n",
    "    elif(bc[0] == 1):\n",
    "        u2[int(2*(bc[2]-1)+1)] = bc[1]\n",
    "\n",
    "# Now impose essential BC's by condensing matrix, and solve for unknown displacements\n",
    "F = F - Kglobal@u2  # compute forces caused by essential boundary conditions\n",
    "mask = np.full(2*(Nele+1),True)\n",
    "for bc in BCs:\n",
    "    if(bc[0]==0):\n",
    "        mask[int(2*(bc[2]-1))] = False\n",
    "    elif(bc[0]==1):\n",
    "        mask[int(2*(bc[2]-1)+1)] = False\n",
    "u_new = np.linalg.solve(Kglobal[mask].T[mask].T,F[mask])\n",
    "\n",
    "# assemble the new displacements into the overall displacement array\n",
    "u2[mask] = u_new\n",
    "\n",
    "# assemble internal forces based on individual element stiffness matrices\n",
    "F_beam = np.zeros(4*Nele)\n",
    "F_py = np.zeros(4*Nele)\n",
    "I = 0\n",
    "for i in range(Nele):\n",
    "    Kbeam = get_Kbeam(EI,dx2)\n",
    "    F_beam[2*I:2*I+4] = Kbeam@u2[I:I+4]\n",
    "    Kpy = get_Kpy_variable(kyiH,b,x2_ele[2*i],x2_ele[2*i+1],H,n)\n",
    "    F_py[2*I:2*I+4] = Kpy@u2[I:I+4]\n",
    "    I+=2\n",
    "fig, ax = plt.subplots(ncols=5,sharey='row',figsize=(8,4))\n",
    "ax[0].plot(u2[0::2],x2)\n",
    "ax[1].plot(u2[1::2],x2)\n",
    "ax[2].plot(F_beam[1::2]*-ele_sign,x2_ele)\n",
    "ax[3].plot(F_beam[0::2]*ele_sign,x2_ele)\n",
    "ax[4].plot(-2*F_py[0::2]/dx2,x2_ele)\n",
    "ax[0].invert_yaxis()\n",
    "ax[0].set_ylabel('depth (m)')\n",
    "ax[0].set_xlabel('u (m)')\n",
    "ax[1].set_xlabel(r'$\\theta$')\n",
    "ax[2].set_xlabel('M (kN-m)')\n",
    "ax[3].set_xlabel('V (kN)')\n",
    "ax[4].set_xlabel('p (kN/m)')\n",
    "for a in ax:\n",
    "    a.grid(True, alpha=0.5)\n",
    "plt.tight_layout()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}