{ "metadata": { "name": "", "signature": "sha256:8153891fd31915fc027b9835e482fe359e89a7a1a683b21a2334c8771f72e58d" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter4-Stress-Strain Relations" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Ex1-pg113" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "## initialization of variables\n", "import numpy\n", "E=2.*10**6. ## kg/cm^2\n", "G=8.*10**5. ## kg/cm^2\n", "ep=numpy.matrix([[0.001, 0, -0.002],\n", " [0 ,-0.003, 0.0005],\n", " [-0.002, 0.0005, 0]])\n", "## calculations\n", "nu=E/(2.*G)-1.\n", "D=E*nu/((1.+nu)*(1.-2.*nu))\n", "mu=G\n", "sigma=2.*mu*ep[0,0]+D*(ep[0,0]+ep[1,1]+ep[2,2])\n", "sigma=2.*mu*ep[1,1]+D*(ep[0,0]+ep[1,1]+ep[2,2])\n", "sigma=2.*mu*ep[2,2]+D*(ep[0,0]+ep[1,1]+ep[2,2])\n", "tau=2.*mu*ep[0,1]\n", "tau=2.*mu*ep[0,2]\n", "tau=2.*mu*ep[1,2]\n", "tau=numpy.matrix([[sigma, tau, tau],\n", " [tau, sigma, tau],\n", " [tau, tau, sigma]])\n", "## results\n", "print'%s %.2f %s %.2f %s'%('The lames constants are ',D,' and ',mu,'kg/cm^2')\n", "print('\\n The stres tensor is')\n", "print(tau)\n", "print('in text book calculations are done wrong')" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The lames constants are 800000.00 and 800000.00 kg/cm^2\n", "\n", " The stres tensor is\n", "[[-1600. 800. 800.]\n", " [ 800. -1600. 800.]\n", " [ 800. 800. -1600.]]\n", "in text book calculations are donw wrong\n" ] } ], "prompt_number": 18 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Ex2-pg114" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "## initialization of variables\n", "#find the magnitude and direction of all the principal strains\n", "sigma_x=1000. ##kg/cm^2\n", "sigma_y=-500. ##kg/cm^2\n", "sigma_z=0. ##kg/cm^2\n", "tau_xy=500. ##kg/cm^2\n", "E=2.*10**6 ## kg/cm^2\n", "nu=0.25\n", "##calculations\n", "ep_x=1./E*(sigma_x-nu*(sigma_y+sigma_z))\n", "ep_y=1./E*(sigma_y-nu*(sigma_x+sigma_z))\n", "ep_z=1./E*(sigma_z-nu*(sigma_y+sigma_x))\n", "J1=ep_x+ep_y+ep_z\n", "sigma_1=(sigma_x+sigma_y)/2.+math.sqrt((1/2.*(sigma_x-sigma_y))**2+tau_xy**2)\n", "sigma_2=(sigma_x+sigma_y)/2.-math.sqrt((1/2.*(sigma_x-sigma_y))**2+tau_xy**2)\n", "th=1/2.*math.atan(2.*tau_xy/(sigma_x-sigma_y))\n", "th=th*180/math.pi\n", "ep_1=1./E*(sigma_1-nu*sigma_2)\n", "ep_2=1./E*(sigma_2-nu*sigma_1)\n", "ep_3=-1./E*nu*(sigma_1+sigma_2)\n", "##results\n", "print'%s %.5f %s'%('The magnitude of principal strain are ',abs(ep_1),'')\n", "print'%s %.5f %s'%('The magnitude of principal strain are ',abs(ep_2),'')\n", "print'%s %.5f %s'%('The magnitude of principal strain are ',abs(ep_3),'')\n", "print'%s %.2f %s'%('\\n and the diection is given by theta=',th,' degrees')\n", "print'%s %.8f %s'%('\\n J1 is ',J1,'')\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The magnitude of principal strain are 0.00066 \n", "The magnitude of principal strain are 0.00047 \n", "The magnitude of principal strain are 0.00006 \n", "\n", " and the diection is given by theta= 16.85 degrees\n", "\n", " J1 is 0.00012500 \n" ] } ], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Ex3-pg115" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "#findd the value of sigma y and principal stress\n", "# initialization of variables\n", "\n", "sigma_x=1400. ##kg/cm^2\n", "tau_xy=400.## kg/cm^2\n", "ep_z=-3.6*10**-6\n", "nu=1/4.\n", "E=2*10**8 ##kg/cm^2\n", "## calculations\n", "sigma_y=(-ep_z*E/nu)-sigma_x\n", "sigma_1=(sigma_x+sigma_y)/2.+math.sqrt((1/2.*(sigma_x-sigma_y))**2+tau_xy**2)\n", "sigma_2=(sigma_x+sigma_y)/2-math.sqrt((1/2.*(sigma_x-sigma_y))**2+tau_xy**2)\n", "th=0.5*math.atan(2*tau_xy/(sigma_x-sigma_y))\n", "th=th*180/math.pi\n", "print'%s %.2f %s'%('sigma_y is ',sigma_y,' kg/cm^2')\n", "print'%s %.2f %s %.2f %s '%('\\n The principal stresses are',sigma_1,'kg/cm^2 'and '',sigma_2,'kg/cm^2')\n", "print'%s %.2f %s'%('\\n The direction is given by theta = ',-th,' degrees')\n", "\n", "## angle was given wrong in the text\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "sigma_y is 1480.00 kg/cm^2\n", "\n", " The principal stresses are 1842.00 1038.00 kg/cm^2 \n", "\n", " The direction is given by theta = 42.14 degrees\n" ] } ], "prompt_number": 3 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Ex4-pg121" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "##initialization of variables\n", "#detemine wheather there is yielding according to tresca and von moises conditions or not\n", "C=1000./3. ##kg/cm^2\n", "sigma_x=2.*C\n", "sigma_y=4.*C\n", "tau_xy=4.*C\n", "sigma_0=4.*C\n", "sigma_1=3.+C*math.sqrt(2.)\n", "sigma_2=3.-C*math.sqrt(2.)\n", "sigma_3=0.\n", "tau_oct=1/3.*math.sqrt((sigma_1-sigma_2)**2+(sigma_2-sigma_3)**2+(sigma_3-sigma_1)**2)\n", "tau_max=sigma_1/2.\n", "taU=1.885*C\n", "tau_y=2.*C\n", "print'%s %.2f %s'%('Actual tau is ',taU,'')\n", "print'%s %.2f %s'%('\\n tau_max at yield is ',tau_y,'')\n", "print('\\n Hence yielding doesn not occur according to Von-Miles condition \\n but it occurs due to Tresca condition')\n", "print('\\n In text book C is not multiplied' )\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Actual tau is 628.33 \n", "\n", " tau_max at yield is 666.67 \n", "\n", " Hence yielding doesn not occur according to Von-Miles condition \n", " but it occurs due to Tresca condition\n", "\n", " In text book C is not multiplied\n" ] } ], "prompt_number": 5 } ], "metadata": {} } ] }