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diff --git a/Solid_Mechanics/Chapter4.ipynb b/Solid_Mechanics/Chapter4.ipynb new file mode 100755 index 00000000..27418162 --- /dev/null +++ b/Solid_Mechanics/Chapter4.ipynb @@ -0,0 +1,242 @@ +{
+ "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": {}
+ }
+ ]
+}
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