Break-Off-Concrete_PB.ipynb

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   "source": [
    "# 1.0 General\n",
    "## 1.1 Project Information\n",
    "**Project Number:** 15M-00685-13  \n",
    "**Project Name:** New Champlain Bridge Corridor Project  \n",
    "**Project Details:**  NCR pending (Formwork displaced for P6A) Axis 1 Diaphragm)  \n",
    "**Project Initiation Date:** June 19th, 2015  \n",
    "**Engineer:** MG  \n",
    "**Checker:**  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "## 1.2 Calculation Description\n",
    "When pouring the end diaphragm at Axis 1 of P6A, the formwork displaced so that the step for the approach slab is not above reinforcement.  But it is possible that the concrete has sufficient resistance even without\n",
    "reinforcement in this area."
   ]
  },
  {
   "cell_type": "markdown",
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    }
   },
   "source": [
    "## 1.3 References\n",
    "References (§, Fig., Tab.) are to CAN/CSA-S6-06 and CAN/CSA-S6.1-06 unless noted otherwise."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "## 1.4 Assumptions:\n",
    "1. Cracks are straight, starting at a distance $a$ from the free edge and\n",
    "   occurring at an angle $θ$ relative to vertical.\n",
    "2. Horizontal force pulling the approach slab away from the end diaphragm is\n",
    "   resisted by the reinforcing dowels.",
    "\n",
    "![Image of end diaphragm](images/Approach-slab-seat.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "# 2.0 Setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
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   },
   "outputs": [],
   "source": [
    "from math import pi, sqrt, sin, cos\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "#plt.rc('text', usetex=True)\n",
    "#plt.rc('font', family='serif')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
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    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "numpy 1.16.2\n",
      "Fri May 29 2020 \n",
      "\n",
      "CPython 3.7.3\n",
      "IPython 5.8.0\n",
      "\n",
      "compiler   : GCC 8.3.0\n",
      "system     : Linux\n",
      "release    : 4.19.0-8-amd64\n",
      "machine    : x86_64\n",
      "processor  : \n",
      "CPU cores  : 2\n",
      "interpreter: 64bit\n",
      "Git hash   :\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -n -v -m -g -iv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
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     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unit calculation and physics extensions activated.\n"
     ]
    }
   ],
   "source": [
    "%load_ext physics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "# 3.0 Inputs\n",
    "## 3.1 Load Factors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
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   },
   "outputs": [],
   "source": [
    "α_L = 1.70                          # Tab. 3.1\n",
    "α_D2 = 1.25\n",
    "α_D3 = 1.50"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "## 3.2 Material Properties"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
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   },
   "outputs": [],
   "source": [
    "ϕ_c = 0.75                      # GREEK PHI SYMBOL ϕ, not Φ or φ\n",
    "fprime_c = 25 MPa\n",
    "rho_c = 2400 kg/m^3\n",
    "γ_c = (25 kN/m^3)\n",
    "γ_asphalt = (23.5 kN/m^3)\n",
    "c = 1.00 MPa\n",
    "μ = 1.40"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "## 3.3 Applied Loading"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The factored wheel load is 223.125 kN\n",
      "The vertical shear is 231.56591 kN/m\n"
     ]
    }
   ],
   "source": [
    "P_wheel = (87.5 kN)                 # load of one wheel Fig. 3.2\n",
    "DLA = 0.50                          # §3.8.4.5.3\n",
    "P_f = α_L * P_wheel * (1 + DLA) ; print('The factored wheel load is %s' % P_f)\n",
    "q_fL =  P_f / ( (600 mm) + 2 * (250 mm))\n",
    "q_fD = ( α_D2 * 0.5 * (6 m) * (0.25 m) * γ_c\n",
    "        + α_D3 * 0.5 * Q(6, 'm') * Q(0.05, 'm') * γ_asphalt )\n",
    "q_f = q_fL + q_fD\n",
    "q_f.convert('kN/m') ; print('The vertical shear is %s' % q_f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "# 4.0 ULS\n",
    "## 4.1 Force Over Crack"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
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     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Factored shear stress on the crack is 1.1578295 MPa\n"
     ]
    }
   ],
   "source": [
    "def v_f(a, θ, q_f):\n",
    "    cracklength = a / np.sin(θ)\n",
    "    shearstress = q_f * np.cos(θ) / cracklength\n",
    "    shearstress.convert('MPa')\n",
    "    return shearstress\n",
    "\n",
    "print('Factored shear stress on the crack is %s' % v_f((100 mm), pi / 4.0, q_f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "## 4.2 Unit Shear Resistance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "def shearfriction(σ):\n",
    "    shearfriction_1 = ϕ_c * (c + μ * σ)\n",
    "    # if shearfriction_1.value < 6.5 :\n",
    "    #     shearfriction = shearfriction_1\n",
    "    # else:\n",
    "    #     shearfriction = (6.5 MPa)\n",
    "    shearfriction = shearfriction_1\n",
    "    return shearfriction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "## 4.3 Resistance Over Crack"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Resistance to shear stress on the crack is 1.965721 MPa\n"
     ]
    }
   ],
   "source": [
    "def v_r(a, θ, q_f):\n",
    "    cracklength = a / np.sin(θ)\n",
    "    normalstress = q_f * np.sin(θ) / cracklength\n",
    "    shearresistance = shearfriction(normalstress)\n",
    "    return shearresistance\n",
    "\n",
    "print('Resistance to shear stress on the crack is %s' % v_r((100 mm), pi / 4.0, q_f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "# 5.0 Results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "autoscroll": false,
    "ein.hycell": false,
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   },
   "outputs": [
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "θ = np.linspace(0.1, pi/2.1, 100)\n",
    "\n",
    "plt.plot(np.degrees(θ), v_f((150 mm), θ, q_f) / v_r((150 mm), θ, q_f), label='a=150 mm')\n",
    "plt.plot(np.degrees(θ), v_f((100 mm), θ, q_f) / v_r((100 mm), θ, q_f), label='a=100 mm')\n",
    "plt.plot(np.degrees(θ), v_f((50 mm), θ, q_f) / v_r((50 mm), θ, q_f), label='a=50 mm')\n",
    "plt.plot(np.degrees(θ), v_f((25 mm), θ, q_f) / v_r((25 mm), θ, q_f), label='a=25 mm')\n",
    "\n",
    "plt.xlabel('θ')\n",
    "plt.ylabel('v_f / v_r')\n",
    "plt.title(\"Cracking\")\n",
    "\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "# 6.0 Conclusions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "The applied load will likely break off the edge of the step that supports the approach slab, at about 50 mm from the edge, and at an angle of 20° from vertical; however, in order for the approach slab to be\n",
    "unsupported, it would have to break off at 150 mm from\n",
    "the edge, for which there is more than enough esistance.\n",
    "\n",
    "The step for the approach slab is acceptable even though it is not above reinforced concrete."
   ]
  }
 ],
 "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": "3.7.3"
  },
  "name": "Break-Off-Concrete_PB.ipynb"
 },
 "nbformat": 4,
 "nbformat_minor": 2
}