{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Examples 7 -- Internal Forces \n", "\n", "Contents:\n", "- [A. ](#A)\n", "- [B. ](#B)\n", "- [C. ](#C)\n", "- [D. ](#D) \n", "- [E. ](#E) \n", "- [F. ](#F) \n", "- [G. ](#G) \n", "- [H. ](#H) \n", "- [I.](#I) \n", "- [J.](#J) \n", "- [K.](#K) " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from numpy import *\n", "dtr = pi/180 # degree-to-radian conversion\n", "from matplotlib.pyplot import *\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## A. Internal forces at points along shaft\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "X1,X2,X3 = 4.,161.,97. # mm (left to right)\n", "F = 71.7 # N" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer at Point A:\n", " NA= 0.0 N\n", " VA= 71.69999999999982 N\n", " MA= -6.954900000000023 N-m\n", " NB= 0.0 N\n", " VB= 71.69999999999982 N\n", " MB= -18.498600000000003 N-m\n", " NC= 0.0 N\n", " VC= -4624.650000000001 N\n", " MC= -18.498600000000003 N-m\n" ] } ], "source": [ "Dy = F*(X2+X3)/X1\n", "Ey = F + Dy\n", "NA = 0.\n", "VA = -Dy + Ey\n", "MA = Ey*X2 - Dy*(X1+X2)\n", "NB = 0.\n", "VB = -Dy + Ey\n", "MB = -Dy*X1\n", "NC = 0.\n", "VC = -Dy\n", "MC = -Dy*X1\n", "print ('answer at Point A:')\n", "print (' NA=',NA,'N')\n", "print (' VA=',VA,'N')\n", "print (' MA=',MA/1e3,'N-m')\n", "print (' NB=',NB,'N')\n", "print (' VB=',VB,'N')\n", "print (' MB=',MB/1e3,'N-m')\n", "print (' NC=',NC,'N')\n", "print (' VC=',VC,'N')\n", "print (' MC=',MC/1e3,'N-m')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "## B.\n", "\n", "(page 466)\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "T = 101. # lb\n", "width = 0.5 # in.\n", "XIJ = 1.40 # in.\n", "X_notch = 0.85 # in.\n", "XGnotch = 2.85 # in.\n", "YED = 1.4 # in.\n", "if 0: # book parameters\n", " T = 100. # lb\n", " width = 0.5 # in.\n", " XIJ = 1.50 # in.\n", " X_notch = 1.0 # in.\n", " XGnotch = 3.0 # in.\n", " YED = 1.5 # in." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " ND= 0.0 lb\n", " VD= 101.0 lb\n", " MD= 0.0 in-lb\n", " NE= 0.0 lb\n", " VE= 101.0 lb\n", " ME= -141.39999999999998 in-lb\n", " NF= -71.4177848998413 lb\n", " VF= 71.4177848998413 lb\n", " MF= -223.53045263481746 in-lb\n" ] } ], "source": [ "MD = ND = 0.\n", "VD = T\n", "NE = 0.\n", "VE = T\n", "ME = -YED*T\n", "NF = -T/sqrt(2)\n", "VF = T/sqrt(2)\n", "r = YED # outer radius of turn\n", "w = width\n", "YEF = (r-0.5*w)/sqrt(2)\n", "MF = -T*(YED+YEF)\n", "print ('answer:')\n", "print (' ND=',ND,'lb')\n", "print (' VD=',VD,'lb')\n", "print (' MD=',MD,'in-lb')\n", "print (' NE=',NE,'lb')\n", "print (' VE=',VE,'lb')\n", "print (' ME=',ME,'in-lb')\n", "print (' NF=',NF,'lb')\n", "print (' VF=',VF,'lb')\n", "print (' MF=',MF,'in-lb')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "\n", "## C.\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "W = 3.69 # kN\n", "XCD = 1.60 # m" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " NL= -3.69 kN\n", " VL= -3.69 kN\n", " ML= -5.904 kN-m\n", " NO= -3.69 kN\n", " VO= -3.69 kN\n", " MO= 0.0 kN-m\n" ] } ], "source": [ "MO = 0.\n", "NO = VO = -W\n", "NL = VL = -W\n", "ML = -W*XCD\n", "print ('answer:')\n", "print (' NL=',NL,'kN')\n", "print (' VL=',VL,'kN')\n", "print (' ML=',ML,'kN-m')\n", "print (' NO=',NO,'kN')\n", "print (' VO=',VO,'kN')\n", "print (' MO=',MO,'kN-m')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "\n", "## D.\n", "\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "P = 6. # kN\n", "L = 3. # m" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " V= 6.0 kN, M= 6.0 x kN\n", " V= 0 kN, M= 6.0 kN\n", " V= -6.0 kN, M= 18.0 - 6.0 x kN\n" ] } ], "source": [ "print ('answer:')\n", "print (' V=',P,'kN, M=',P,'x kN')\n", "print (' V=',0,'kN, M=',P*L/3,' kN')\n", "print (' V=',-P,'kN, M=',P*L,'-',P,'x kN')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "## E\n", "\n", "\n", "\n", "For the cantilever beam shown, determine the shear and moment as a function of position and graph these functions." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "X = 5.68 # m\n", "w = 7.76 # kN/m" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " V=( -7.76 kN/m) x m\n", " M=( -3.88 kN/m) x^2 m^2\n" ] } ], "source": [ "print ('answer:')\n", "print (' V=(',-w,'kN/m) x m')\n", "print (' M=(',-w/2,'kN/m) x^2 m^2')" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$V(x)$ (lb)')" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x = arange(0.,X,0.1)\n", "v = -w*x\n", "%matplotlib inline\n", "grid()\n", "plot( x, v, color='black')\n", "xlabel('$x$ (ft)')\n", "ylabel('$V(x)$ (lb)')" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$M(x)$ (ft-lb)')" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "m = -w/2*x**2\n", "%matplotlib inline\n", "grid()\n", "plot( x, m, color='blue')\n", "xlabel('$x$ (ft)')\n", "ylabel('$M(x)$ (ft-lb)')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "checked; works." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "## F. Beam with overhang\n", "\n", "\n", "\n", "A beam with an overhang is subjected to the uniformly distributed load shown. Determine (and graph) the shear and moment as functions of position." ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [], "source": [ "w = 530. # lb/ft\n", "X = 7.75 # ft (half length)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " for =0" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x = arange(0.,2*X,0.1)\n", "N = int(len(x)/2)\n", "v1 = -w*x[:N]\n", "v2 = By - w*x[N:]\n", "%matplotlib inline\n", "grid()\n", "plot( x[:N], v1, label= '$0" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "m1 = -w/2*x[:N]**2\n", "m2 = -By*X + By*x[N:] -w/2*x[N:]**2\n", "%matplotlib inline\n", "grid()\n", "plot( x[:N], m1, label= '$0\n", "\n", "\n", "## G. Loaded Beam\n", "\n", "\n", "\n", "Determine (and graph) the shear and moment as functions of position." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "w0 = 8.73 # kN/m\n", "L = 5.08 # m" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " V= -0.859251968503937 x^2\n", " M= -0.28641732283464566 x^3\n" ] } ], "source": [ "F0 = w0/(2*L) # times x^2\n", "XF = 2./3. # times x\n", "print( 'answer:')\n", "print( ' V=',-F0,'x^2')\n", "print (' M=',-F0/3,'x^3')" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$V(x)$ (kN)')" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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GA1sB5wGfmtl3iW25mX2X2vJEJBMUFob5GzfeCOPHQ9eu8PnnsauSGJIZ0yhw93ru3tDdNy23NXT3TdNRpIhkhlNOgccfD63V99xTLdbzUdJtRMyspILnDqpuAWY2IDG7fK2ZdVrvtRFmtsDM5plZ7+p+lohU34EHwvPPh8l/e+0V1iSX/FGZ3lO3m1n7nx6Y2VGES1bV9TZwKPB8+SfNrC0wEGgH9AFuUs8rkcxQUgIvvxw65vbuDffdF7siSZfKhMbhwN1mtpOZnQCcAvSqbgHuPtfd51Xw0sHAaHdf6e4fAguA0up+nojUjG22gRdegH32gcGD4dJL1bMqH5hX4qdsZjsAjwGLgP7u/n2NFWI2Ffiru89MPL4BeNnd70s8/hfwlLv/Yr0xMxsKDAUoLi4uGT16dJXrKCsro6ioqMrvzzb5drygY65pq1YZl1++I889V0z//p/wpz/NpzADrgno51w53bp1m+XunTa230bbiJjZW4QZ4D9pDBQCr5gZ7r5LEv/Gs0CTCl46190f39DbKniuwoRz99sIEw/p1KmTd+3adWMlbdDUqVOpzvuzTb4dL+iYU6FHDzj7bLjqqmYUFjbjvvtgk01S9nFJ0c85NZLpPXVgdT/E3XtW4W2Lga3LPW4OfFrdWkSk5hUUwJVXwlZbwbBhsGxZmNux+eaxK5OalkxoLPSNXMMyM9vYPlUwDviPmV1DmCfSGphRw58hIjXojDOgaVP4wx/CeuQTJ0Lz5rGrkpqU1HoaZnaambUo/6SZ1TGz7mZ2N3BMVQsws0PMbDHQGRhvZpMA3H0O8BDwDjARONXd1cBAJMMNHAhPPQULF4Zmh5rLkVuSCY0+wBrgATP71MzeMbMPgPnAUcD/uftdVS3A3ce6e3N3r+vuxe7eu9xrl7j7du7ext2fqupniEh69egR5nKsXh3OOF55JXZFUlOSmRH+g7vf5O57A9sAPQgNC7dx9xPcfXbKqxSRrNOhQ2h22KhRCJFnnoldkdSEyszTwN1Xuftn7v5NqgoSkdyx3XYhOLbdFg44IKwMKNmtUqEhIlJZTZvCtGlQWhoWd7r11tgVSXUoNEQk5TbfPKz+17cvnHSSZo9ns0qHhpk1UA8oEams+vXD3I2jj4Zzz4Uzz1RwZKNkZoQXEBoHHg3sDqwE6prZMmACcJu7z09plSKSE2rXhnvuCYPjV18Ny5fDTTeREW1HJDnJTO6bAjwLjADedve1EJaABboBl5nZ2J96RImI/JqCAhg1KqwIeNllUFYGd90VAkUyXzKh0dPdV63/pLt/BYwBxpiZftwikjQz+Oc/YdNN4W9/C8vHPvgg1K0buzLZmGTmaawCMLNrzayiJoJUFCoiIhszYkRYd/zxx+Ggg7T2eDaozEB4GTDOzBoAmFkvM3shNWWJSL447TS480547jno0we+/TZ2RfJrkrk8BYC7n2dmg4CpZrYSWAGck7LKRCRvDBkCRUUwaBD07AmTJkHjxrGrkopUZo3wHsAJhLDYEvizu/83VYWJSH4ZMADGjoU33wxtR774InZFUpHKXJ46Fzjf3bsSln590My6p6QqEclLBx4I48bBu+9Ct26wZEnsimR9SYeGu3d39+mJ798C+gL/SFVhIpKfeveG8ePhgw+ga1f4VEuvZZSNhsav3DH1GaHj7Qb3ERGpiu7dwwJOixfDvvvCokWxK5KfVGsRJqBzdRdhEhGpSJcuoV/V0qUhOD76KHZFAsmFximsW4Tps5pehElEZEM6d4Znn4VvvgmXqhQc8SVzy+2TwFRCQHwGbAF8rzU1RCQddt89BEfPniE4pk6Fli0jF5XHkjnT2BF4DZgGXAGsUmCISDp17BiC47vvdMYRWzJtRH509xuAnYBPgBlmdpGZNUx5dSIiCQqOzFCZW25/cPergPbAD8BrZvbXlFUmIrIeBUd8lZkR3tLM+gB/BFoAy4FLU1WYiEhFFBxxJTNP400z+wp4DDgWaARMJtxmW5TS6kREKlA+OLp1g4ULY1eUP5K5e+oQ4AN3LcwoIpmjY0d45pnQp6p7d5g2DZo1i11V7ktmIPx9BYaIZKKSktARd+nSEB6ffx67otxXmYaFIiIZZ489YMKE0HKkRw9Ytix2RblNoSEiWW+ffeCJJ0KTw5494csvY1eUuxQaIpITunULy8bOmwe9ekFZWdJrzEklKDREJGf06gWPPgpvvQVnnbULy5fHrij3KDREJKfsvz889BDMm9eQfv3g++9jV5RbFBoiknP694dzzpnLtGlw+OHw44+xK8odCg0RyUn77beUW24Jd1YdfTSsXh27otygkSIRyVlDh0JZGQwfDg0awJ13QoH+VK4WhYaI5LRhw0JwXHghFBXBqFGgBaqrTqEhIjnv/PNh+XK46qoQHJddFrui7KXQEJGcZwZXXBHOOC6/HBo3hrPOil1VdlJoiEheMIMbbgjrjZ99Nmy+OZxwQuyqsk/0ISEzG2Bmc8xsrZl1Kvd8SzP73sxmJ7ZbYtYpItmvsBDuvhv69oWTToJHHoldUfbJhDONt4FDgVsreO19d++Q5npEJIfVqRPColcvGDQINtsM9tsvdlXZI/qZhrvPdfd5sesQkfxRvz48+SS0bRsmAr78cuyKsodlylIZZjYV+Ku7z0w8bgnMAd4DvgPOc/f/buC9Q4GhAMXFxSWjR4+uch1lZWUUFeXPgoT5drygY84XyRzzV1/V4bTTdmP58lpcd91sWrVakabqUqM6P+du3brNcvdOG93R3VO+Ac8SLkOtvx1cbp+pQKdyj+sCv0l8XwIsAjbd2GeVlJR4dUyZMqVa7882+Xa87jrmfJHsMX/wgftWW7k3ber+4YcpLSnlqvNzBmZ6Er/P0zKm4e49q/CelcDKxPezzOx9YAdgZg2XJyJ5rFUrePrpsCZH794wfTpsuWXsqjJX9DGNDTGzLc2sMPH9tkBr4IO4VYlILmrXLoxxLFwIBxwQ5nNIxaKHhpkdYmaLgc7AeDOblHjpd8CbZvYG8Ahwkrt/FatOEclte+8dWqq/9lrojLtqVeyKMlP00HD3se7e3N3runuxu/dOPD/G3du5+67u3tHdn4hdq4jktoMOgltvhUmT4LjjYO3a2BVlnkyYpyEikjGOPx6WLIFzz4Xi4tCvStZRaIiIrGfECPj8c7j66hAcZ54Zu6LModAQEVmPGVx7LSxdGhobbrVVWMhJFBoiIhUqKAh9qpYsgSFDoEkT6NEjdlXxRR8IFxHJVHXrwtix0KYNHHoovPlm7IriU2iIiPyKRo3COuMNG8L++8OiRbErikuhISKyEVtvDU89FVb/69s3rMmRrxQaIiJJaN8+XKp6773QGXflytgVxaHQEBFJUvfucNddMG0aHHNMfk7+091TIiKVMGgQLF4cloxt2RIuuyx2Reml0BARqaQzz4QPP4TLL4fttsuvtcYVGiIilWQGo0bBxx/DySdDixahrXo+0JiGiEgV1KoFDz4IO+8MAwbkzxwOhYaISBU1bBjW4dh007AOx6efxq4o9RQaIiLV0Lw5jB8f5m4ccECYy5HLFBoiItW0665hAae33oKBA2H16tgVpY5CQ0SkBvTtCzfeGFqODBsWu5rU0d1TIiI15MQTYf78sA5HmzZw6qmxK6p5Cg0RkRp0+eUhOP785zCHo0+f2BXVLF2eEhGpQYWFcP/9oVfVkUfCnDmxK6pZCg0RkRpWVARPPAENGsCBB4YVAHOFQkNEJAW23hrGjQsr//XvDz/8ELuimqHQEBFJkU6d4N574aWX4PjjwT12RdWn0BARSaHDDoNLL4X//Af+8Y/Y1VSf7p4SEUmxc86BuXPhggugXbuw3ni20pmGiEiKmcFtt8Gee8LgwTB7duyKqk6hISKSBptsEpaLbdwY+vULA+TZSKEhIpImTZqEO6q+/BIOOSQ71xlXaIiIpNFuu8Hdd4c7qoYOzb47qhQaIiJpdvjhMHIk3HNP6FOVTRQaIiIRnH9+WPHvrLNCZ9xsodAQEYmgoADuugs6dICjjoJ582JXlByFhohIJPXrw2OPQd26cPDB8O23sSvaOIWGiEhELVrAI4/A++/D0UfDmjWxK/p1Cg0Rkch+9zu4/vqw1vgFF8Su5tepjYiISAY46aQwU/zSS8Oa40ccEbuiiulMQ0QkA5jBqFGw994wZEjmthqJHhpmdqWZvWtmb5rZWDNrVO61EWa2wMzmmVnvmHWKiKRanTowZkxoNdK/PyxbFruiX4oeGsAzwM7uvgvwHjACwMzaAgOBdkAf4CYzK4xWpYhIGhQXhx5VS5bAwIGwenXsin4uemi4+9Pu/tN/lpeB5onvDwZGu/tKd/8QWACUxqhRRCSdOnWCW2+FyZNDW/VMkmkD4ccBDya+b0YIkZ8sTjz3C2Y2FBgKUFxczNSpU6tcQFlZWbXen23y7XhBx5wvsv2YW7SAQw7Znquvbk79+u/QvfvGFxpPyzG7e8o34Fng7Qq2g8vtcy4wFrDE4xuB35d7/V/AYRv7rJKSEq+OKVOmVOv92Sbfjtddx5wvcuGYf/zRfZ993OvVc3/jjY3vX51jBmZ6Er/P03Km4e49f+11MzsGOBDokSgewpnF1uV2aw58mpoKRUQyT+3a8PDDUFISWqm/+moYJI8p+piGmfUBzgb6ufv/yr00DhhoZnXNrBXQGpgRo0YRkViaNAkzxhctyowZ49FDA7gBaAg8Y2azzewWAHefAzwEvANMBE519wyfYC8iUvM6dw5zOCZOhAsvjFtL9IFwd9/+V167BLgkjeWIiGSkoUNh5ky45JJ1l6tiyIQzDRER2QgzuOEGKC2FY46B996LU4dCQ0QkS9StGwbG69SBww6DFSvSX4NCQ0Qki7RoAQ88AHPmxFljXKEhIpJl9tsPLr4Y/vOfcMkqnRQaIiJZaMQIOOggGDYMXnwxfZ+r0BARyUIFBXDPPeFy1YABocFhWj43PR8jIiI1rVGj0Er9q69CR9w1ayzln6nQEBHJYh06wC23wNSpcMcdrVL+edEn94mISPUccwy89hoUFn6f8s/SmYaISA647jro1++zlH+OQkNERJKm0BARkaQpNEREJGkKDRERSZpCQ0REkqbQEBGRpCk0REQkaQoNERFJmnm6m7GnmJktAz6uxj+xBfBFDZWTDfLteEHHnC90zJWzjbtvubGdci40qsvMZrp7p9h1pEu+HS/omPOFjjk1dHlKRESSptAQEZGkKTR+6bbYBaRZvh0v6JjzhY45BTSmISIiSdOZhoiIJE2hkWBmfcxsnpktMLNzYteTamZ2p5ktNbO3Y9eSLma2tZlNMbO5ZjbHzE6PXVOqmdkmZjbDzN5IHPPfY9eUDmZWaGavm9mTsWtJFzP7yMzeMrPZZjYzZZ+jy1Phf2DAe8B+wGLgVeAod38namEpZGa/A8qAe9x959j1pIOZNQWauvtrZtYQmAX0z/GfswEN3L3MzGoD04HT3f3lyKWllJkNAzoBm7r7gbHrSQcz+wjo5O4pnZuiM42gFFjg7h+4+4/AaODgyDWllLs/D3wVu450cvfP3P21xPfLgblAs7hVpZYHZYmHtRNbTv+laGbNgQOAO2LXkosUGkEzYFG5x4vJ8V8m+c7MWgK7Aa/ErST1EpdqZgNLgWfcPdeP+VrgLGBt7ELSzIGnzWyWmQ1N1YcoNAKr4Lmc/mssn5lZETAG+Iu7fxe7nlRz9zXu3gFoDpSaWc5ejjSzA4Gl7j4rdi0R7O3uHYG+wKmJS9A1TqERLAa2Lve4OfBppFokhRLX9ccA97v7o7HrSSd3/waYCvSJXEoq7Q30S1zfHw10N7P74paUHu7+aeLrUmAs4bJ7jVNoBK8Crc2slZnVAQYC4yLXJDUsMSj8L2Cuu18Tu550MLMtzaxR4vt6QE/g3bhVpY67j3D35u7ekvD/48nu/vvIZaWcmTVI3NyBmTUAegEpuTNSoQG4+2rgT8AkwuDoQ+4+J25VqWVmDwAvAW3MbLGZHR+7pjTYGxhM+OtzdmLbP3ZRKdYUmGJmbxL+OHrG3fPmNtQ8UgxMN7M3gBnAeHefmIoP0i23IiKSNJ1piIhI0hQaIiKSNIWGiIgkTaEhIiJJU2iIiEjSFBrIGl4KAAABvElEQVQiIpI0hYaIiCRNoSGSYmZWz8ymJVrwJ/ueOmb2vJnVSmVtIpWl0BBJveOAR919TbJvSLTofw44MmVViVSBQkOkGhIrAe6X+P4fZnZ9BbsdDTye2Kelmb1rZneY2dtmdr+Z9TSzF8xsvpmVbzL3WOK9IhlDbUREqiHRfvoi4HZgENCv/BlFogHmQndvknjcElhAWMtjDqEf1BvA8UA/YIi790/sWwh87u5bput4RDZG10tFqsHdn090zx0GdK3gEtQWwDfrPfehu78FYGZzgOfc3c3sLaBluX97jZn9aGYNEysNikSny1Mi1WBm7QmdZFdu4Bf798Am6z23stz3a8s9Xssv/5CrC/xQA6WK1AiFhkgVmVlT4H7CevIrzKz3+vu4+9dAoZmtHxzJ/Pu/AZa5+6pqFytSQxQaIlVgZvWBR4Hh7j4XuBgYuYHdnwb2qcLHdAMmVKlAkRTRQLhIipnZbsAwdx9cyfc9Coxw93mpqUyk8nSmIZJi7v46YfW8Sk3uAx5TYEim0ZmGiIgkTWcaIiKSNIWGiIgkTaEhIiJJU2iIiEjSFBoiIpI0hYaIiCRNoSEiIkn7f8NbuwkqEJEGAAAAAElFTkSuQmCC\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x = arange(0.,L,0.1)\n", "v = -F0*x**2\n", "grid()\n", "plot( x, v, color='blue')\n", "xlabel('$x$ (m)')\n", "ylabel('$V(x)$ (kN)')" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$M(x)$ (kN-m)')" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "m = -F0/3.*x**3\n", "grid()\n", "plot( x, m, color='blue')\n", "xlabel('$x$ (m)')\n", "ylabel('$M(x)$ (kN-m)')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "checked; works." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "## H. Loaded Beam\n", "\n", "\n", "\n", "Determine (and graph) the shear and moment as functions of position." ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [], "source": [ "w1 = 9.75 # kN/m\n", "w2 = 3.53 # kN/m\n", "L = 8.90 # m" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [], "source": [ "F1 = 0.5*L*(w1-w2) # triangle force dist.\n", "X1 = L/3.\n", "F2 = L*w2 # uniform force dist.\n", "X2 = L/2\n", "Ay = F1 + F2\n", "MA = F1*X1 + F2*X2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$V(x)=A_y-\\int_0^x w_2\\; dx'-\\int_0^x \\left (w_1-w_2 \\right )\\left (1-\\frac{x'}{L} \\right )\\; dx'$\n", "\n", "$V(x)=A_y-w_2 x - \\left (w_1-w_2 \\right ) \\left (x- \\frac{x^2}{2L} \\right )$\n", "\n", "$V(x)=A_y-w_1x+ \\frac{w_1-w_2}{2L} x^2$\n", "\n", "$M(x)=-M_A + \\int_0^x \\left (A_y-w_1x'+ \\frac{w_1-w_2}{2L} x'^2 \\right )\\; dx'$\n", "\n", "$M(x)=-M_A + A_yx-\\frac{w_1}{2}x^2+ \\frac{w_1-w_2}{6L} x^3 $" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " V(x) = 59.10 -9.75 x +0.3494 x^2\n", " M(x) = -221.92 +59.10 x -4.88 x^2 +0.1165 x^3\n" ] } ], "source": [ "v0 = Ay\n", "v1 = -w1\n", "v2 = (w1-w2)/(2*L)\n", "m0 = -MA\n", "m1 = Ay\n", "m2 = -w1/2\n", "m3 = (w1-w2)/(6*L)\n", "print ('answer:')\n", "print (' V(x) = {:.2f} {:+.2f} x {:+.4f} x^2'.format(v0,v1,v2))\n", "print (' M(x) = {:.2f} {:+.2f} x {:+.2f} x^2 {:+.4f} x^3'.format(m0,m1,m2,m3))" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$V(x)$ (kN)')" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x = arange(0.,L,0.1)\n", "v = v0 + v1*x + v2*x**2\n", "grid()\n", "plot( x, v, color='blue')\n", "xlabel('$x$ (m)')\n", "ylabel('$V(x)$ (kN)')" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$M(x)$ (kN-m)')" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "m = m0 + m1*x + m2*x**2 + m3*x**3\n", "grid()\n", "plot( x, m, color='blue')\n", "xlabel('$x$ (m)')\n", "ylabel('$M(x)$ (kN-m)')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "checked; works." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## I.\n", "\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [], "source": [ "w1 = 370. # lb/ft\n", "w2 = 740. # lb/ft\n", "L = 18. # ft" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [], "source": [ "F1 = 0.5*L*(w2-w1) # triangle force dist.\n", "X1 = 2*L/3.\n", "F2 = L*w1 # uniform force dist.\n", "X2 = L/2\n", "By = (F1*X1 + F2*X2)/L\n", "Ay = F1+F2-By" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$w(x) = w_1 + (w_2-w_1)\\frac{x}{L} $\n", "\n", "$V(x)=A_y-\\int_0^x w_1\\; dx'-\\int_0^x \\left (w_2-w_1 \\right )\\frac{x'}{L} \\; dx' $\n", "\n", "$V(x)= A_y - w_1 x - \\frac{w_2-w_1 }{2L}x^2 $\n", "\n", "$M(x)= \\int_0^x \\left [ A_y - w_1 x' - \\left (w_2-w_1 \\right ) \\frac{x'^2}{2L} \\right ]\\;dx'$\n", "\n", "$ M(x)= A_yx-\\frac{w_1}{2}x^2+ \\frac{w_1-w_2}{6L} x^3$" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "answer:\n", " V(x) = 4440.00 -370.00 x -10.28 x^2\n", " M(x) = 0.00 +4440.00 x -185.00 x^2 -3.43 x^3\n" ] } ], "source": [ "v0 = Ay\n", "v1 = -w1\n", "v2 = (w1-w2)/(2*L)\n", "m0 = 0.\n", "m1 = Ay\n", "m2 = -w1/2\n", "m3 = (w1-w2)/(6*L)\n", "print ('answer:')\n", "print (' V(x) = {:.2f} {:+.2f} x {:+.2f} x^2'.format(v0,v1,v2))\n", "print (' M(x) = {:.2f} {:+.2f} x {:+.2f} x^2 {:+.2f} x^3'.format(m0,m1,m2,m3))" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$V(x)$ (kN)')" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x = arange(0.,L,0.1)\n", "v = v0 + v1*x + v2*x**2\n", "grid()\n", "plot( x, v, color='blue')\n", "xlabel('$x$ (m)')\n", "ylabel('$V(x)$ (kN)')" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$M(x)$ (ft-lb)')" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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m1c2sOnA/0A5oDnRK9gW4I3muZsAKoFuqv42UaskSuOCCMC3HLbfETiOycTvvDLfeGgZgjhkTO03lkpMC4+4vAtkuXtoeGO3u69z9A2AB0DK5LXD3he7+DTAaaG9mBhwL/DV5/EigQ4X+ApKV9etD75yvvgqTWNbSeaQUiJ494eCDQ9PuF1/ETlN5xL702svMzgVmAFe4+wqgMfBqxj6Lk20AH22w/RBgR+ALd/+uhP1/xMy6A90BGjRowLRp08oVfM2aNeV+bC7lMufjjzdhypSf0rfvPJYuXbrJkwrqmFa8QsmaDzkvvHAbevQoomvXJfTpM7/U/fIhazbyIqe75+QG7ArMzvi5AVCdcBZ1KzA82X4/cE7GfsOA04GOwCMZ27sAg4GfEM5sirc3BWZlk6moqMjLa+rUqeV+bC7lKudbb7nXquXevr37+vXlew4d04pXKFnzJWfv3u5m7tOnl75PvmTdmDRzAjM8i/fYaL3I3P0Td//e3dcDDxOawCCcgTTN2LUJsKSM7Z8BdcysxgbbJUfWrg0rBu64Y1ih0ix2IpHy6d8/jPS/6KKwtIRsnmgFxswaZfz4S6C4h9l44Cwz28LMdgOaAa8D/wKaJT3GahE6AoxPqulU4Izk8V0BLSuUQ1ddFSazHDkS6tWLnUak/LbdFgYNgpkzw8zfsnly1U35L8B0YC8zW2xm3YA7zWyWmc0EWgF9ANz9HeBxYA7wPHBJcqbzHdALmAjMBR5P9gW4GuhrZgsI12SG5eL3EpgyBe6/P1wcPeGE2GlENl+HDnDKKXDjjfDhh7HTFLacXOR3904lbC61CLj7rYTrMhtunwBMKGH7Qv7XxCY5snJlGKi2995w222x04hUDDMYPBiaNw9rFz39dOxEhUsj+aXc+vaFjz+GESNgq61ipxGpOLvsAtdfD+PGweTJsdMULhUYKZdnn4Xhw8PCTYccEjuNSMXr0wf22COcxeiCf/lscoExs9rJqHqpopYvDysD7rsv3HBD7DQi6dhiCxg4MHRguf/+2GkK00YLjJlVM7OzzexvZrYMeBdYambvmNldZtYs/ZiSTy69FD77DEaNCv8JRSqrk0+GNm3CBf9ly2KnKTzZnMFMBfYArgEauntTd68PHEUYcX+7mZ2TYkbJI08+GaaBuf56OOCA2GlE0mUWVr9cuxauuy52msKTTS+y4939Ry2Q7r4ceBJ40sy0ynoV8Nln0KMHFBWFFSpFqoK99w7Lfg8cGGZcluxt9AymuLiY2ZZm1tfMnjKzJ82sj5ltmbmPVG59+oSJAP/4R6ipjxRShdxwA/zkJ6F5WFP6Z29TLvKPIkyhPxi4D9gH+FMaoST/TJgAjz4azlz23Td2GpHc2n57uP12mD4dpkxpEDtOwdiUArOXu3dz96nJrTuwZ1rBJH+sXh2aBpo3h2uvjZ1GJI6uXcOU/g89tDtr18ZOUxg2pcC8ZWaHFv9gZocAr1R8JMk311wDixfDsGHqNSZVV7VqcPfd8NlnW3DPPbHTFIZsuikXzxd2CPBPM1tkZh8Q5hY7Ou2AEtdLL4UxAJdfDoceuvH9RSqzo4+Gww//jNtvV7flbGTTi+zk1FNIXvr667D88a67avljkWIXXbSQ88+vx803hznLpHQbLTDurvlEq6ibb4b33gtrldeuHTuNSH7Yeee1XHghDB0aepXtqSvRpcqmiWy1ma0q4bbazFblIqTk3qxZcOed8Otfaxp+kQ3ddFO4HqnxYGXLZhzMtu6+XQm3bd19u1yElNxavz6s6FenDvzhD7HTiOSfBg3CQntPPQWvqKtTqco1m7KZNazoIJI/Hnkk9Pe/++6wDLKI/NgVV0CjRnDllRp8WZryTtf/o0W/pHJYtgyuvhqOOQa6dImdRiR/1a4N/fuHD2NPPRU7TX4qb4GxCk0heeOKK+DLL+GBB8JEfyJSuvPOgxYtwrWY776LnSb/lLfAPFyhKSQvvPBCmA6mX78wwZ+IlK16dbj1Vpg/H0aOjJ0m/2RdYMzsjuLv3X3IhtuksH39dZgpeY89NB2MyKY49VRo2RJ+9ztYty52mvyyKWcwJXVWbVdRQSSuO+4In8IeeAC23DJ2GpHCYRbOYj76CB58MHaa/JLNOJgeZjYL2MvMZmbcPgBmph9R0rZgAdx2G3TqpDEvIuVx3HGhY8ytt4ZrmBJkcwZzImG6mOrAKRm3InfXSpYFzj0sprTFFqFbsohsuuKzmGXLNH1MpmwKzB7J13nAKmB1csPMdkgpl+TIM8/Ac8+F9uNGjWKnESlchx8OJ54YZsD44ovYafJDNgVmKPA8sBfwxga3GelFk7R99RX07h3WeenVK3YakcJ3yy2wYgUMGBA7SX7IZqqYQe6+D/BHd9/d3XfLuO2eg4ySkrvugg8+gPvu0xLIIhXhwAOhY0cYOBA+/TR2mviyuchvAO7eY2P7SOH44AP4/e/hV7+CVq1ipxGpPPr3h7VrwxLLVV02TWRTzexSM9s5c6OZ1TKzY81sJNA1nXiSlr59wwp9msxSpGLtvXeYZmnIEPjvf2OniSubAtMW+B74i5ktMbM5ZrYQmA90Aga6+4gUM0oFe/55ePppuP56aNIkdhqRyue3v4VvvtEHuGwWHPsaGAIMMbOaQD3gK3dXP4kCtG5d6Ja8557Qp0/sNCKV009/Cp07h4HLV10F9evHThTHJs1F5u7fuvtSFZfCNXhwGLF/771h7IuIpOO660JPzarco6y8k11KAVq2LCyDfNJJ0LZt7DQildtee8FZZ4Vemp99FjtNHCowVcgNN4TeLVW9XVgkV667Lvyfu+ee2Eni2OQCY2a1zax6GmEkPbNmwcMPQ8+emopfJFdatIAzzoBBg8IAzKomm3Ew1czsbDP7m5ktA94FlprZO2Z2l5k1Sz+mbA73cEF/++3hxhtjpxGpWn77W1i9Olz3rGqyGgdDmI/sGqChuzd19/rAUcCrwO1mVuakl2Y23MyWmdnsjG07mNlkM5uffK2bbDczG2RmC5JZmw/KeEzXZP/5ZtY1Y3uRmc1KHjNIAz9/6Nlnw2JiN90EO2j2OJGc2m8/+OUvQzPZypWx0+RWNgXmeHe/2d1nuvv64o3uvtzdn3T304ExG3mOEYTxNJn6AS+4ezPgheRnCGvMNEtu3YEH4P8n1rwROARoCdxYXJSSfbpnPE6XsBPffmv85jfhgmOPUudiEJE03XBDKC5VbablbOYi+xbAzO4p7cygeJ8ynuNFYPkGm9sDxYuMjgQ6ZGwf5cGrQB0zawS0ASYnhW0FMBlom9y3nbtPd3cHRmU8V5U3blxj3nsvdJXUfGMicRxwQFj5csCA0FxWVWx0oGWGNcB4MzvL3b80s9bAje5+RDlfu4G7LwVw96VmVjwUqTHwUcZ+i5NtZW1fXML2EplZd8LZDg0aNGDatGnlCr9mzZpyPzZXVq6swYgRLTn44OVstdVM8jxuQRxTKJycUDhZCyUnlD9r27bbMn58EVdfvYAzz1y88Qdsprw4pu6e9Q04G/gX8DIwEThqEx67KzA74+cvNrh/RfL1b8CRGdtfAIqAK4HfZmy/HrgC+DkwJWP7UcAz2WQqKiry8po6dWq5H5srvXq5V6u23mfPjp0kO4VwTN0LJ6d74WQtlJzum5e1VSv3xo3d162ruDylSfOYAjM8i/fYrLspm9lxwIXAl8BPgMvc/aVyVzb4JGneIvm6LNm+GGiasV8TYMlGtjcpYXuVNmdOmKbilFOW0KJF7DQiAtCvH3z8MTz2WOwkubEp42CuA65392OAM4AxZnbsZrz2eP43C3NXYFzG9nOT3mSHAis9NKVNBFqbWd3k4n5rYGJy32ozOzS5RnRuxnNVWb/5DWyzDZx33qLYUUQkccIJYc2YO+6A9es3vn+hy7rAuPux7v5y8v0sQm+vW7J5rJn9BZgO7GVmi82sG3A7cIKZzQdOSH4GmAAsBBYADwM9k9dcDtxMaKL7F9A/2QbQA3gkecz7wHPZ/l6V0ZQpYRnk66+H7bcvs/+FiOSQGVx9NcybB+OqwMfgjV7kNzNL2tx+wMOF+ePK2idj306l3HVcCfs6cEkpzzMcGF7C9hnAz0p7/apk/fowe+suu4RlkKdPj51IRDKdfjrssUdYkKxDh1B0KqvNWnAMOEwLjuWX0aPhrbfC2uCaLVkk/9SoAVdeCa+/Tt737Nxcm7rg2NJkwbEP0IJjeWfdujC53gEHwNlnx04jIqXp2hUaNKj8yypnMw7mcOAJd9eCY3nugQdg0SKYODEshywi+WnLLcP8gP36hRaHAw+MnSgd2bwNTQFmmdkSwhiVK4D2ZnagmakRJk+sXBmaxY47LvRUEZH8dvHFsN12oUdZZZVNgbmMMK5kEKHX2DzCwMcBwIfpRZNNceed8Pnn4Y+1Ml80FKkstt8+zA/4xBPw/vux06Qjm7nI7gOOABy4B/gWuNzdW7l7w5TzSRY+/hgGDoROnaCoKHYaEcnWZZdB9ephvZjKKKuWenf/yt3vAI4Bfgq8bmaHpBlMsnfTTfDdd3DrrbGTiMim2Gmn8MFw2DD4ohJe1c5mwbGjzKy7mQ0gTMvfjjBdzI5ph5ONmzMHhg8PK1XutlvsNCKyqfr0gS+/DCvOVjbZnMH8gzBS/r9AD3cvcvdj3H1CutEkG9dcA7Vrh1XzRKTwHHAAHHtsaCb7tpJNvJFNgekBvAKcBLyWjIMZY2a/NTOtuxLRyy/D+PFh6ol69WKnEZHy6tsXFi+Gv/41dpKKlc1F/gfdvZe7/8LdGxAmmRxJuNh/etoBpWTuobA0agS9e8dOIyKbo127sOrsgAHh/3ZlsSkLjgHg7osJU+SriSyiCRPgn/+EoUNDE5mIFK5q1cIHxR494JVX4MgjYyeqGBrvXYDWrw8zJe++O5x/fuw0IlIRzj0XdtghnMVUFiowBeipp8L0EjfeCDVrxk4jIhVh663DGczTT1eegZcqMAXm++/hhhtgn32gc+fYaUSkIl1ySZht+d57YyepGCowBebPf4a5c6F//zACWEQqj0aNwkzow4fDihWx02w+FZgC8u23YdT+gQfCaafFTiMiaSgeeDn8R0srFh4VmAIyfDgsXAg336zp+EUqq/33h6OOgiFDQpN4IdPbVIH4+utQWA47DE48MXYaEUnTpZeGD5PPPRc7yeZRgSkQQ4eGWZNvvVXT8YtUdh06QOPGMHhw7CSbRwWmAKxZA7fdFhYTa9UqdhoRSVvNmmFBskmTYN682GnKTwWmAAweDJ9+GlasFJGqoXt3qFUL7r8/dpLyU4HJc198EVarPPlkOPTQ2GlEJFfq14czz4QRI2D16thpykcFJs8NGBCKzM03x04iIrl26aWhuIwaFTtJ+ajA5LHly8OI3jPOCGtGiEjV0rJluN13X2HOsqwCk8fuuQdWrQpTw4hI1dSrF7z7LkyZEjvJplOByVMrVoSzl9NPh333jZ1GRGI588xwPea++2In2XQqMHnq3nt19iIisMUWoUfZM8/ABx/ETrNpVGDy0BdfhOaxX/4S9tsvdhoRie2ii8L0UA89FDvJplGByUP33gsrV+rsRUSCJk3glFNg2DD45pvYabKnApNnVq4MZy8dOqjnmIj8z8UXhwHXY8fGTpI9FZg8M2hQaCLT2YuIZDrhBNhtN3jwwdhJsqcCk0dWrgwDK089Naz5IiJSrFq1cLF/6tTQbbkQqMDkkcGDw9nLjTfGTiIi+ei888JEmIVysT96gTGzRWY2y8zeNrMZybYdzGyymc1PvtZNtpuZDTKzBWY208wOyniersn+882sa6zfp7xWrQpnL6ecAgcdtPH9RaTqadAgrGY7YgR89VXsNBsXvcAkWrn7Ae5+cPJzP+AFd28GvJD8DNAOaJbcugMPQChIwI3AIUBL4MbiolQoBg8Ogyt19iIiZbnoovBe8cQTsZNsXL4UmA21B0Ym348EOmRsH+XBq0AdM2sEtAEmu/tyd18BTAba5jp0ea1eHc5eTjoJiopipxGRfHbMMbDnnoVxsT8fCowDk8zsDTPrnmxr4O5LAZKv9ZPtjYGPMh67ONlW2vaCcP/9YWJLnb2IyMaYhS7L//wnzJwZO03ZasQOABzh7ksNfJJvAAAN30lEQVTMrD4w2czK6h9R0mLBXsb2Hz9BKGLdARo0aMC0adM2MW6wZs2acj8209dfV+OOOw7l5z9fw5dfzqQCnvIHKipnLhRK1kLJCYWTtVByQn5k3WOPGtSseTg33LCU3r3nl7hPPuTE3fPmBtwE/AaYBzRKtjUC5iXfPwh0yth/XnJ/J+DBjO0/2K+0W1FRkZfX1KlTy/3YTIMHu4P7P/5RIU/3IxWVMxcKJWuh5HQvnKyFktM9f7J26eK+7bbuq1eXfH+aOYEZnsV7etQmMjOrbWbbFn8PtAZmA+OB4p5gXYFxyffjgXOT3mSHAis9NKFNBFqbWd3k4n7rZFte++absFrlkUfC0UfHTiMiheTii8P129GjYycpXewmsgbAWDMrzvJnd3/ezP4FPG5m3YD/AB2T/ScAJwILgLXAeQDuvtzMbgb+lezX392X5+7XKJ/HHoOPPiqMi3Uikl8OOwyaN4dHHoELLoidpmRRC4y7LwT2L2H758BxJWx34JJSnms4MLyiM6bl++/h9tvDiP22BdPfTUTyhVkoLH37wjvvQIsWsRP9WD70IquSnnoK3nsPrr02/KGIiGyqLl3CyP5hw2InKZkKTATucOutsNdeYc0XEZHyqFcP2reHUaNg3brYaX5MBSaC556Df/8b+vWD6tVjpxGRQnbBBfD55zB+fOwkP6YCk2PFZy877wydO8dOIyKF7vjjoWnT/GwmU4HJsRdfDCNwr7oqtJ2KiGyO6tXDLMuTJsGHH8ZO80MqMDl2221Qvz6cf37sJCJSWZx3Xvg6YkTUGD+iApNDM2aETxl9+8JWW8VOIyKVxa67hqayP/4R1q+PneZ/VGBy6Pe/hzp1oEeP2ElEpLLp1i00kb3wQuwk/6MCkyNz54axL5deCtttFzuNiFQ2HTrADjuEkf35QgUmR+66KzSLXXZZ7CQiUhltsQWccw48/XTotpwPVGByYMkSePTRcApbr17sNCJSWXXrFibRffTR2EkCFZgcuPfeMPdY376xk4hIZbbffnDwwfnTm0wFJmUrV8LQodCxI+y2W+w0IlLZde0Kb78N779fO3YUFZi0PfQQrFoFV14ZO4mIVAWdOoVB3BMnNowdRQUmTevWwT33wHHHQVFR7DQiUhXsuCOcfDJMmdKAb7+Nm0UFJkV//nO4wH/VVbGTiEhV0rUrrFhRi4mR1/VVgUnJ+vWha/L++8MJJ8ROIyJVSbt2sP323zByZNwcKjApmTAhDK686iotKCYiuVWrFhx33DLGj4flERePV4FJyZ13hin5O3aMnUREqqI2bf7LN9/AmDHxMqjApGD6dHjppTDuRVPyi0gMzZqtYd99idpMpgKTgrvugrp1w6haEZEYzMLF/tdeg3ffjZNBBaaCzZsX5gK65BLYZpvYaUSkKuvcOSxIFussRgWmgt19d7jAdumlsZOISFXXsCG0aQN/+lOYrirXVGAq0CefwKhR8Otfh1UrRURi69oVPv4Y/v733L+2CkwFeuCBMHpfk1qKSL449dSw0GGMZjIVmAry1VcwZAiccgrsuWfsNCIiwZZbwplnwtixsGZNbl9bBaaCPPYYfPop9OkTO4mIyA917gxr18K4cbl9XRWYCuAOAwfCAQfAMcfETiMi8kNHHhkGfud6ITIVmAowaRLMmROuvWhaGBHJN9WqhbOYSZNCZ6ScvW7uXqryGjAAGjWCX/0qdhIRkZKdc06YhHf06Ny9pgrMZpo9O3wq6NUrjH8REclHzZvDgQfmtplMBWYz3XMPbLUVXHRR7CQiImU75xyYMSPMOJILKjCb4ZNPwqeBrl3DKnIiIvnsrLPC9ZjHHsvN66nAbIbigZW9e8dOIiKycTvtFJZwf/TR0Ps1bSow5fTNN9UYMiSsfb3XXrHTiIhk55xzwmzvy5al/1qVqsCYWVszm2dmC8ysX5qvNXlyfQ2sFJGC06ULvPEGNGiQ/mtVmgJjZtWB+4F2QHOgk5k1T+O13OGvf23K/vtDq1ZpvIKISDpyOVavRu5eKnUtgQXuvhDAzEYD7YE5Ff1CkyfDokW1GTlSAytFREpTac5ggMbARxk/L062VbgBA2CHHdZx1llpPLuISOVgnouuBDlgZh2BNu5+QfJzF6Clu1+6wX7dge4ADRo0KBq9icNa16+HBx/cg+22W0Xnzp9WTPgUrVmzhm0KZGnNQslaKDmhcLIWSk4onKxp5mzVqtUb7n7wRnd090pxAw4DJmb8fA1wTVmPKSoq8vKaOnVquR+bS4WS071wshZKTvfCyVooOd0LJ2uaOYEZnsX7cmVqIvsX0MzMdjOzWsBZwPjImUREqqxKc5Hf3b8zs17ARKA6MNzd34kcS0Skyqo0BQbA3ScAE2LnEBGRytWLTERE8ogKjIiIpEIFRkREUqECIyIiqVCBERGRVFSakfzlYWafAh+W8+H1gM8qME5aCiUnFE7WQskJhZO1UHJC4WRNM+cu7v6Tje1UpQvM5jCzGZ7NVAmRFUpOKJyshZITCidroeSEwsmaDznVRCYiIqlQgRERkVSowJTfQ7EDZKlQckLhZC2UnFA4WQslJxRO1ug5dQ1GRERSoTMYERFJhQrMRphZWzObZ2YLzKxfCfdvYWZjkvtfM7NdI2RsamZTzWyumb1jZpeXsM8xZrbSzN5ObjfkOmdGlkVmNivJMaOE+83MBiXHdKaZHRQh414Zx+ptM1tlZr032CfaMTWz4Wa2zMxmZ2zbwcwmm9n85GvdUh7bNdlnvpl1jZDzLjN7N/m3HWtmdUp5bJl/JznKepOZfZzxb3xiKY8t830iBznHZGRcZGZvl/LYnB7T6AuF5fONMO3/+8DuQC3g30DzDfbpCQxNvj8LGBMhZyPgoOT7bYH3Ssh5DPBs7GOaZFkE1Cvj/hOB5wADDgVey4O/g/8S+v7nxTEFjgYOAmZnbLsT6Jd83w+4o4TH7QAsTL7WTb6vm+OcrYEayfd3lJQzm7+THGW9CfhNFn8fZb5PpJ1zg/vvBm7Ih2OqM5iytQQWuPtCd/8GGA2032Cf9sDI5Pu/AseZmeUwI+6+1N3fTL5fDcwFGucyQwVrD4zy4FWgjpk1ipjnOOB9dy/voNwK5+4vAss32Jz5tzgS6FDCQ9sAk919ubuvACYDbXOZ090nuft3yY+vAk3Sev1NUcoxzUY27xMVpqycyXvPmcBf0nr9TaECU7bGwEcZPy/mx2/c/79P8p9mJbBjTtKVIGmiOxB4rYS7DzOzf5vZc2bWIqfBfsiBSWb2hpl1L+H+bI57Lp1F6f9h8+WYAjRw96UQPnQA9UvYJ9+O7fmEs9WSbOzvJFd6Jc15w0tpdsynY3oU8Im7zy/l/pweUxWYspV0JrJht7ts9skJM9sGeBLo7e6rNrj7TUITz/7AYODpXOfLcIS7HwS0Ay4xs6M3uD+fjmkt4FTgiRLuzqdjmq18OrbXAd8Bj5Wyy8b+TnLhAWAP4ABgKaH5aUN5c0yBTpR99pLTY6oCU7bFQNOMn5sAS0rbx8xqANtTvtPszWJmNQnF5TF3f2rD+919lbuvSb6fANQ0s3o5jlmcZUnydRkwltDEkCmb454r7YA33f2TDe/Ip2Oa+KS4KTH5uqyEffLi2CadC04GOntycWBDWfydpM7dP3H37919PfBwKRny5ZjWAE4DxpS2T66PqQpM2f4FNDOz3ZJPsmcB4zfYZzxQ3BPnDODvpf2HSUvS7joMmOvuA0rZp2HxtSEza0n4t/88dyn/P0dtM9u2+HvCBd/ZG+w2Hjg36U12KLCyuOknglI/EebLMc2Q+bfYFRhXwj4TgdZmVjdp7mmdbMsZM2sLXA2c6u5rS9knm7+T1G1w7e+XpWTI5n0iF44H3nX3xSXdGeWY5qo3QaHeCD2a3iP0Erku2daf8J8DYEtC88kC4HVg9wgZjyScks8E3k5uJwIXAxcn+/QC3iH0cHkVODzS8dw9yfDvJE/xMc3MasD9yTGfBRwcKevWhIKxfca2vDimhKK3FPiW8Am6G+Ha3wvA/OTrDsm+BwOPZDz2/OTvdQFwXoScCwjXLIr/Vot7Ye4ETCjr7yRC1j8lf4MzCUWj0YZZk59/9D6Ry5zJ9hHFf5sZ+0Y9phrJLyIiqVATmYiIpEIFRkREUqECIyIiqVCBERGRVKjAiIhIKlRgREQkFSowIiKSChUYkTxiZluZ2T/MrHry82UW1vn5m5n1zNivlpm9mEwPIpKXNNBSJI+Y2SWEtVLuTX5+lzAfmhPWnvlZxr43EqaJL22ySJGodAYjkiMWVh09Ifn+FjMbVMJunUnmEDOzoYTpPcYD84A9kpUI70r2fTrZXyQv6QxGJEeSqdH7E2blPZswn933GffXAv7j7g0zti0izCW2DT8+g6kO/Nfdf5Kb30Bk06j9ViRH3P3FZPblvsAxmcUlUQ/4YhOe73sz+8bMtvWwkqlIXlETmUiOmNm+QCNgXSkF4SvC7NybYgvg683NJpIGFRiRHEjWFXmMsFb7l2bWZsN93H0FUN3MSioyq4FtN3jOHYFP3f3bFCKLbDYVGJGUmdnWwFPAFe4+F7gZuKmU3ScR1vf5AXf/HHjFzGZnXORvBUyo+MQiFUMX+UXyiJkdCPR19y5Z7PsUcI27z0s/mcim0xmMSB5x97eAqcUDLUuT9Dh7WsVF8pnOYEREJBU6gxERkVSowIiISCpUYEREJBUqMCIikgoVGBERSYUKjIiIpEIFRkREUvF/rua+8htddx8AAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "m = m0 + m1*x + m2*x**2 + m3*x**3\n", "grid()\n", "plot( x, m, color='blue')\n", "xlabel('$x$ (ft)')\n", "ylabel('$M(x)$ (ft-lb)')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "checked; works." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## J.\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "FBD:\n", "\n", "\n", "\n", "Find $F_B$ and $F_C$ from the FBD of the truck:\n", "\n", "$$\n", "\\begin{matrix}\n", "\\sum M_B = 0 & \\Rightarrow & F_C (l_C-l_B) - W (l_E-l_B) =0 & \\Rightarrow & F_C = \\frac{l_E-l_B}{l_C-l_B}W \\\\ \n", "\\sum F_y = 0 & \\Rightarrow & F_B+F_C-W = 0 & \\Rightarrow & F_B = W-F_C \n", "\\end{matrix}\n", "$$\n", "\n", "Construct the load function (using the [Dirac Delta](https://en.wikipedia.org/wiki/Dirac_delta_function)):\n", "\n", "$$w(x) = w_0 + F_B \\; \\delta(x-l_B) + F_C \\; \\delta(x-l_C)$$\n", "\n", "Then integrate to find $V(x)$:\n", "\n", "$$V(x) = \\int_{0}^{x}-w(x')dx'=\\left\\{\\begin{matrix}\n", "-w_0 x, & 0" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# graph functions:\n", "x1 = linspace(0.,lB,100)\n", "x2 = linspace(lB,lC,100)\n", "x3 = linspace(lC, L,100)\n", "V1 = -w0*x1\n", "V2 = -w0*x2 - FB\n", "V3 = -w0*x3 - FB - FC\n", "grid()\n", "linestyle = {'color':'red', 'linewidth':2}\n", "plot( x1, V1, **linestyle)\n", "plot( [x1[-1],x2[0]], [V1[-1],V2[0]], **linestyle) # vertical connecting line\n", "plot( x2, V2, **linestyle)\n", "plot( [x2[-1],x3[0]], [V2[-1],V3[0]], **linestyle) # vertical connecting line\n", "plot( x3, V3, **linestyle)\n", "xlabel('$x$ (m)')\n", "ylabel('$V(x)$ (kN)')" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$M(x)$ (kN-m)')" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "M1 = -0.5*w0*x1**2\n", "M2 = -0.5*w0*x2**2 - FB*x2 + FB*lB\n", "M3 = -0.5*w0*x3**2 - (FB+FC)*x3 + FB*lB + FC*lC\n", "grid()\n", "plot( x1, M1, **linestyle)\n", "plot( x2, M2, **linestyle)\n", "plot( x3, M3, **linestyle)\n", "xlabel('$x$ (m)')\n", "ylabel('$M(x)$ (kN-m)')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## K.\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "w1 = 4.8 # kN/m\n", "L = 2.55 *2 # m (total length)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$w(x) = w_1\\frac{x}{L} + A_y\\delta(x) + B_y\\delta(x-\\frac{L}{2}) $\n", "\n", "where $\\delta(x)$ is the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function).\n", "\n", "$V(x)= Ay - \\frac{w_1}{2L}x^2+By H(x-\\frac{L}{2})$\n", "\n", "where $H(x)$ is the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function).\n", "\n", "$M(x)=\\left\\{\\begin{matrix}\n", "A_yx-\\frac{w_1}{6L}x^3 & 0" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "v0 = Ay + By\n", "x = arange(L/2,L,0.1)\n", "v = v0 + v1*x + v2*x**2\n", "grid()\n", "plot( x, v, color='blue')\n", "xlabel('$x$ (m)')\n", "ylabel('$V(x)$ (kN)')" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0, 0.5, '$M(x)$ (kN-m)')" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "m = m0 + m1*x + m2*x**2 + m3*x**3 + By*(x-L/2)\n", "grid()\n", "plot( x, m, color='blue')\n", "xlabel('$x$ (m)')\n", "ylabel('$M(x)$ (kN-m)')" ] } ], "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": "3.7.1" } }, "nbformat": 4, "nbformat_minor": 1 }