composite_plate_bending.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bending of a composite plate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## INPUT "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "length = 1.5  # m\n",
    "width = 0.15  # m\n",
    "divs = 40\n",
    "material_E1 = 138.0e9  # Pa\n",
    "material_E2 = 9.0e9  # Pa\n",
    "material_G12 = 6.9e9  # Pa\n",
    "material_nu = 0.28\n",
    "material_t = 1.25e-04  # m\n",
    "\n",
    "# ply orientations, 0deg aligned with FEM X-direction\n",
    "ply_ori = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] \n",
    "\n",
    "# applied force in Z-direction at unsupported end\n",
    "force = 0.1 # N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Shear and bending moment diagram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
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     "text": "SPC reaction force should be = 0.10 N, SPC reaction moment should be = 0.15 Nm\n"
    },
    {
     "output_type": "display_data",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "S = force * np.ones(divs+1)\n",
    "M = [0]\n",
    "for i in range(divs):\n",
    "    M.insert(0, length/divs*force + M[0]) \n",
    "\n",
    "y_location = np.arange(divs+1)*length/divs\n",
    "plt.plot(y_location,S,label=\"S\")\n",
    "plt.plot(y_location,M,label=\"M\")\n",
    "plt.xlabel(\"Y location, m\")\n",
    "plt.legend()\n",
    "print(f\"SPC reaction force should be = {S[0]:.2f} N, SPC reaction moment should be = {M[0]:.2f} Nm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Composite D-matrix calculation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "The plate bending stiffness matrix  D is \n [[39.0119699   0.71239249  0.        ]\n [ 0.71239249  2.54425891  0.        ]\n [ 0.          0.          1.940625  ]]\n"
    }
   ],
   "source": [
    "# Using standard CLT approach\n",
    "\n",
    "# material invariants\n",
    "nu_21 = material_nu * material_E2 / material_E1\n",
    "Q11 = material_E1 / (1 - material_nu * nu_21)\n",
    "Q22 = material_E2 / (1 - material_nu * nu_21)\n",
    "Q12 = (material_nu * material_E2 / (1 - material_nu * nu_21))\n",
    "Q66 = material_G12\n",
    "Q = np.array([[Q11,Q12,0.0],[Q12,Q22,0.0],[0.0,0.0,Q66]])\n",
    "\n",
    "z1 = - material_t*len(ply_ori)/2\n",
    "D = np.zeros([3,3])\n",
    "for ply in ply_ori:\n",
    "    ply_rad = ply*np.pi/180.0\n",
    "    # transformation matrix \n",
    "    T = np.array([\n",
    "        [np.cos(ply_rad)**2, np.sin(ply_rad)**2, 2*np.cos(ply_rad)*np.sin(ply_rad)],\n",
    "        [np.sin(ply_rad)**2, np.cos(ply_rad)**2, -2*np.cos(ply_rad)*np.sin(ply_rad)],\n",
    "        [-np.cos(ply_rad)*np.sin(ply_rad), np.cos(ply_rad)*np.sin(ply_rad), np.cos(ply_rad)**2-np.sin(ply_rad)**2]\n",
    "    ])\n",
    "    Q_bar = np.linalg.inv(T) @ Q @ T.transpose()\n",
    "    D = D + 1/3 * Q_bar * ((z1+material_t)**3-z1**3)\n",
    "    z1 += material_t\n",
    "\n",
    "print(\"The plate bending stiffness matrix  D is \\n\", D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate deflections along the Y-direction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": "Plate deflection at loading point should be = 0.296250 m\n"
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 432x288 with 1 Axes>",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "# invert D matrix \n",
    "D_inv = np.linalg.inv(D)\n",
    "\n",
    "# bending curvature along y-direction\n",
    "curv = []\n",
    "for i in M:\n",
    "      cmat = D_inv @ np.array([0, i/width, 0]) \n",
    "      curv.insert(0, cmat[1]) \n",
    "curv.reverse()\n",
    "curv = np.array(curv)\n",
    "\n",
    "# displacement w and gradient dw at the root are assumed = 0\n",
    "dw = np.zeros(divs+1)\n",
    "w = np.zeros(divs+1)\n",
    "mean_curv = (curv[:-1] + curv[1:]) / 2\n",
    "dw[1:] = np.cumsum(mean_curv * (y_location[1:] - y_location[:-1]))\n",
    "mean_grad = (dw[:-1] + dw[1:]) / 2\n",
    "w[1:] = np.cumsum(mean_grad *  (y_location[1:] - y_location[:-1]))\n",
    "\n",
    "# plot deflection along y\n",
    "plt.plot(y_location,w,label=\"w\")\n",
    "plt.xlabel(\"Y location, m\")\n",
    "plt.ylabel(\"Z-deflection, m\")\n",
    "print(f\"Plate deflection at loading point should be = {w[-1]:.6f} m\")"
   ]
  }
 ]
}