{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 3 (30 points)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def compute_Sg(S, angles=(0,0,0)):\n",
    "    \n",
    "    alpha, beta, gamma = np.radians(angles)\n",
    "    \n",
    "    Rg = np.array([[np.cos(alpha) * np.cos(beta),  \n",
    "                    np.sin(alpha) * np.cos(beta),  \n",
    "                    -np.sin(beta)],\n",
    "                   [np.cos(alpha) * np.sin(beta) * np.sin(gamma) - np.sin(alpha) * np.cos(gamma), \n",
    "                    np.sin(alpha) * np.sin(beta) * np.sin(gamma) + np.cos(alpha) * np.cos(gamma),  \n",
    "                    np.cos(beta) * np.sin(gamma)],\n",
    "                   [np.cos(alpha) * np.sin(beta) * np.cos(gamma) + np.sin(alpha) * np.sin(gamma), \n",
    "                    np.sin(alpha) * np.sin(beta) * np.cos(gamma) - np.cos(alpha) * np.sin(gamma),  \n",
    "                    np.cos(beta) * np.cos(gamma)]])\n",
    "                  \n",
    "    return np.dot(Rg.T, np.dot(S,Rg))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  6.00000000e+01,  -1.83697020e-15,  -3.74939946e-32],\n",
       "       [ -1.83697020e-15,   4.50000000e+01,  -3.06161700e-16],\n",
       "       [ -3.74939946e-32,  -3.06161700e-16,   5.00000000e+01]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = np.diag([60, 50, 45])\n",
    "\n",
    "S_G = compute_Sg(S, angles=(180,0,90)); S_G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def compute_unit_vectors(strike, dip):\n",
    "    \n",
    "    strike = np.radians(strike)\n",
    "    dip = np.radians(dip)\n",
    "    \n",
    "    n = np.array([-np.sin(strike) * np.sin(dip), np.cos(strike) * np.sin(dip), -np.cos(dip) ])\n",
    "    \n",
    "    ns = np.array([ np.cos(strike), np.sin(strike), 0 ])\n",
    "    \n",
    "    nd = np.array([ -np.sin(strike) * np.cos(dip), np.cos(strike) * np.cos(dip), np.sin(dip) ])\n",
    "    \n",
    "    return (n, ns, nd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**i) $strike = 45^{\\circ}, dip = 65^{\\circ}$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.64085638  0.64085638 -0.42261826]\n",
      "[ 0.70710678  0.70710678  0.        ]\n",
      "[-0.29883624  0.29883624  0.90630779]\n"
     ]
    }
   ],
   "source": [
    "n, ns, nd = compute_unit_vectors(45, 65); \n",
    "print(n)\n",
    "print(ns)\n",
    "print(nd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we compute the normal and shear stresses on the plane"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "56.427876096865383"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_G = np.array([[47.5, -12.5, 0],[-12.5, 47.5, 0],[0,0,40]])\n",
    "\n",
    "S_n = np.dot(np.dot(S_G, n), n); S_n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-7.1054273576010019e-15"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau_s = np.dot(np.dot(S_G, n), ns); tau_s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.6604444311897772"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau_d = np.dot(np.dot(S_G, n), nd); tau_d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.6604444311897772"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau_mag = np.sqrt(tau_d ** 2 + tau_s ** 2); tau_mag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "43.660468711549093"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pc = (0.6 * S_n - tau_mag) / 0.6; Pc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}