UnitStress_Composite_Section.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculation of axial load and bending moment that will produce a unit stress"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Axial load"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The stress from an axial load is\n",
    "\n",
    "$$\n",
    "\\sigma_{axial} = \\frac{P}{A}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import pi\n",
    "from pint import UnitRegistry\n",
    "u = UnitRegistry()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "D_leg = 1917.7*u.mm\n",
    "t_leg = 38.1*u.mm\n",
    "d_leg = D_leg - 2*t_leg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "A_leg = pi*(D_leg**2 - d_leg**2)/4.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "D_pile = 2133.6*u.mm\n",
    "t_pile = 38.1*u.mm\n",
    "d_pile = D_pile - 2*t_pile"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "A_pile = pi*(D_pile**2 - d_pile**2)/4.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = A_leg + A_pile"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P = 4.7580e+05 newton\n"
     ]
    }
   ],
   "source": [
    "P=1.0E6*u.Pa*A\n",
    "print('P = {0:0.4e}'.format(P.to(u.N)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bending moment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The composite area moment of inertia of a grouted section can be defined as\n",
    "$$\n",
    "I_{comp} = \\frac{\\pi}{64}\\left[\\left(D_{pile}^4 + D_{leg}^4\\right) - \\left(d_{pile}^4 +d_{leg}^4\\right)\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "I_comp=pi/64.0*((D_pile**4 + D_leg**4) - (d_pile**4 + d_leg**4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The extreme fiber stress from bending is calculated from\n",
    "\n",
    "$$\n",
    "\\sigma_b = \\frac{M c}{I}\n",
    "$$\n",
    "\n",
    "We want to solve for the moment which will produce a unit stress on the extreme fiber"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M = 2.2226e+08 millimeter * newton\n"
     ]
    }
   ],
   "source": [
    "c = D_pile/2\n",
    "M = 1.0e6*u.Pa*I_comp/c\n",
    "print('M = {0:0.4e}'.format(M.to(u.N*u.mm)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Extreme fiber stress = 1.00 megapascal\n"
     ]
    }
   ],
   "source": [
    "print('Extreme fiber stress = {0:0.2f}'.format((M*c/I_comp).to(u.MPa)))"
   ]
  }
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