{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter 10:Strain Transformation" ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.1 Page No 491" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "ep_x = 500 #Normal Strain\n", "ep_y = -300 #Normal Strain\n", "gamma_xy = 200 #Shear Strain\n", "\n", "#Calculation\n", "import math\n", "theta = 30*(math.pi/180)\n", "theta = theta*-1\n", "ep_x_new = ((ep_x+ep_y)/2) + ((ep_x - ep_y)/2)*math.cos(2*theta) + (gamma_xy/2)*math.sin(2*theta)\n", "gamma_xy_new = -((ep_x - ep_y)/2)*math.sin(2*theta) + (gamma_xy/2)*math.cos(2*theta)\n", "gamma_xy_new = 2*gamma_xy_new\n", "phi = 60*(math.pi/180)\n", "ep_y_new = (ep_x+ep_y)/2 + ((ep_x - ep_y)/2)*math.cos(2*phi) + (gamma_xy/2)*math.sin(2*phi)\n", "\n", "#Display\n", "print'The equivalent strain acting on the element oriented at 30 degrees clockwise = ',round(ep_y_new,1),\"*10**-6\"\n", "print'The equivalent shear strain acting on the element = ',round(gamma_xy_new,0),\"10**-6\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The equivalent strain acting on the element in the y plain oriented at 30 degrees clockwise = -13.4 *10**-6\n", "The equivalent shear strain acting on the element = 793.0 10**-6\n" ] } ], "prompt_number": 4 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.2 Page No 492" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "ep_x = -350.0 #(*10**-6) Normal Strain\n", "ep_y = 200.0 #*(10**-6) Normal Strain\n", "gamma_xy = 80.0 #*(10**-6) Shear Strain\n", "\n", "#Calculation\n", "#Orientation of the element\n", "import math\n", "tan_thetap = (gamma_xy)/(ep_x - ep_y)\n", "thetap1 =math.atan(tan_thetap)*180/3.14+180\n", "thetap=thetap1/2.0\n", "\n", "#Principal Strains\n", "k = (ep_x + ep_y)/2\n", "l = (ep_x - ep_y)/2\n", "tou = gamma_xy/2\n", "R = math.sqrt( (l)**2 + tou**2)\n", "ep1 = R + k\n", "ep2 = k -R \n", "ep_x1 = k + l*math.cos(2*-4.14*3.14/180.0)+ tou*math.sin(2*-4.14*3.14/180.0)\n", "\n", "#Display\n", "print'The orientation of the element in the positive counterclockwise direction = ',round(thetap,1),\"degree\"\n", "print'The principal strains are ',round(ep1,0),\"*10**-6 and \",round(ep2,0),\"*10**-6\"\n", "print'The principal strain in the new x direction is ',round(ep_x1,0),\"*10**-6\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The orientation of the element in the positive counterclockwise direction = 85.9 degree\n", "The principal strains are 203.0 *10**-6 and -353.0 *10**-6\n", "The principal strain in the new x direction is -353.0 *10**-6\n" ] } ], "prompt_number": 39 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.3 Page No 493" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "ep_x = -350 #(*10**-6) Normal Strain\n", "ep_y = 200.0 #*(10**-6) Normal Strain\n", "gamma_xy = 80.0 #*(10**-6) Shear Strain\n", "\n", "#Orientation of the element\n", "import math\n", "tan_thetap = -(ep_x - ep_y)/(gamma_xy)\n", "thetap1 = math.atan(tan_thetap)*180/3.14+180\n", "thetap=thetap1/2.0\n", "\n", "#Maximum in-plane shear strain\n", "l = (ep_x - ep_y)/2\n", "tou = gamma_xy/2\n", "R = sqrt( l**2 + tou**2)\n", "max_inplane_strain = 2*R\n", "gamma_xy_1 = (-l*math.sin(2*thetap1)+ tou*math.cos(2*thetap1))*2\n", "strain_avg = (ep_x + ep_y)/2\n", "thetap1 = thetap1*(180/math.pi)\n", "thetap2 = (90 + thetap1)\n", "\n", "#Display\n", "print'The orientation of the element =',round(thetap,0,),\"degre\"\n", "print'The maximum in-plane shear strain = ',round(max_inplane_strain,0),\"*10**-6\"\n", "print'The average strain =',strain_avg,\"*10**-6\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The orientation of the element = 131.0 degre\n", "The maximum in-plane shear strain = 556.0 *10**-6\n", "The average strain = -75.0 *10**-6\n" ] } ], "prompt_number": 42 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.4 Page No 496" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "ep_x = 250.0 #(*10**-6) Normal Strain\n", "ep_y = -150.0 #*(10**-6) Normal Strain\n", "gamma_xy = 120.0 #*(10**-6) Shear Strain\n", "\n", "#Calculation\n", "#Construction of the circle\n", "import math\n", "strain_avg = (ep_x + ep_y)/2\n", "tou = gamma_xy/2\n", "R = sqrt((ep_x - strain_avg)**2 + (tou**2))\n", "#Principal Strains\n", "ep1 = (strain_avg + R)\n", "ep2 = (strain_avg - R)\n", "tan_thetap = (tou)/(ep_x - strain_avg)\n", "thetap1 = (math.atan(tan_thetap))/2.0\n", "thetap1 = thetap1*(180/math.pi)\n", "\n", "#Display\n", "print'The principal strains are = ',round(ep1,0),\"*10**-6 and \",round(ep2,0),\"*10**-6\"\n", "print'The orientation of the element = ',round(thetap1,2),\"degree\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The principal strains are = 259.0 *10**-6 and -159.0 *10**-6\n", "The orientation of the element = 8.35 degree\n" ] } ], "prompt_number": 52 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.5 Page No 497" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "ep_x = 250 #(*10**-6) Normal Strain\n", "ep_y = -150 #*(10**-6) Normal Strain\n", "gamma_xy = 120 #*(10**-6) Shear Strain\n", "\n", "#calculation\n", "#Orientation of the element\n", "thetas = 90 - 2*8.35\n", "thetas1 = thetas/2\n", "#Maximum in-plane shear strain\n", "l = (ep_x - ep_y)/2\n", "tou = gamma_xy/2\n", "R = sqrt( l**2 + tou**2)\n", "max_inplane_strain = 2*R\n", "strain_avg = (ep_x + ep_y)/2\n", "\n", "#Display\n", "print'The orientation of the element ',thetas1,\"degree\"\n", "print'The maximum in-plane shear strain',round(max_inplane_strain,0),\"*10**-6\"\n", "print'The average strain = ',strain_avg,\"*10**-6\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The orientation of the element 36.65 degree\n", "The maximum in-plane shear strain 418.0 *10**-6\n", "The average strain = 50 *10**-6\n" ] } ], "prompt_number": 56 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.6 Page No 498" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "ep_x = -300 #(*10**-6) Normal Strain\n", "ep_y = -100 #*(10**-6) Normal Strain\n", "gamma_xy = 100 #*(10**-6) Shear Strain\n", "theta = 20 #degrees\n", "\n", "#Calculation\n", "#Construction of the circle\n", "import math\n", "strain_avg = (ep_x+ ep_y)/2.0\n", "tou = gamma_xy/2.0\n", "R = sqrt((-ep_x + strain_avg)**2 + tou**2)\n", "#Strains on Inclined Element\n", "theta1 = 2*theta\n", "phi = math.atan((tou)/(-ep_x +strain_avg))\n", "phi = phi*(180/math.pi)\n", "psi = theta1 - phi\n", "psi = psi*(math.pi/180)\n", "ep_x1 = -(-strain_avg+ R*math.cos(psi))\n", "gamma_xy1 = -(R*math.sin(psi))*2\n", "ep_y1 = -(-strain_avg - R*math.cos(psi))\n", "\n", "#Display\n", "print'The normal strain in the new x direction = ',round(ep_x1,0),\"10**-6\"\n", "print'The normal strain in the new y direction = ',round(ep_y1,1),\"10**-6\"\n", "print'The shear strain in the new xy direction = ',round(gamma_xy1,0),\"10**-6\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The normal strain in the new x direction = -309.0 10**-6\n", "The normal strain in the new y direction = -91.3 10**-6\n", "The shear strain in the new xy direction = -52.0 10**-6\n" ] } ], "prompt_number": 62 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.7 Page No 503" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "ep_x = -400 #(*10**-6) Normal Strain\n", "ep_y = 200 #*(10**-6) Normal Strain\n", "gamma_xy = 150 #*(10**-6) Shear Strain\n", "\n", "#calculation\n", "#Maximum in-plane Shear Strain\n", "strain_avg = (ep_x+ ep_y)/2\n", "tou = gamma_xy/2\n", "R = sqrt((-ep_x + strain_avg)**2 + tou**2) \n", "strain_max = strain_avg + R\n", "strain_min = strain_avg - R\n", "max_shear_strain = strain_max - strain_min\n", "#Absolute Maximum Shear Strain\n", "abs_max_shear = max_shear_strain\n", "\n", "#Display\n", "print'The maximum in-plane shear strain= ',round(max_shear_strain,0),\"10**-6\"\n", "print'The absolute maximum shear strain ',round(abs_max_shear,0),\"10**-6\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The maximum in-plane shear strain= 618.0 10**-6\n", "The absolute maximum shear strain 618.0 10**-6\n" ] } ], "prompt_number": 66 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.8 Page No 505" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "import math\n", "ep_a = 60.0 #(*10**-6) Normal Strain\n", "ep_b = 135.0 #*(10**-6) Normal Strain\n", "ep_c = 264.0 #*(10**-6) Normal Strain\n", "theta_a = 0\n", "theta_b = 60*(math.pi/180)\n", "theta_c = 120*(math.pi/180)\n", "\n", "#Calculation\n", "a1 = (math.cos(theta_a))**2\n", "b1 = (math.sin(theta_a))**2\n", "c1 = math.cos(theta_a)*math.sin(theta_a)\n", "a2 = (math.cos(theta_b))**2\n", "b2 = (math.sin(theta_b))**2\n", "c2 = math.cos(theta_b)*math.sin(theta_b)\n", "a3 = (math.cos(theta_c))**2\n", "b3 = (math.sin(theta_c))**2\n", "c3 = math.cos(theta_c)*math.sin(theta_c)\n", "\n", "ep_x = 60 #*10**-6\n", "ep_y = 246 #*10**-6\n", "gamma_xy = -149 #*10**-6\n", "strain_avg = (ep_x + ep_y )/2.0\n", "tou = gamma_xy/2.0\n", "R = sqrt((-ep_x + strain_avg)**2 + tou**2) \n", "ep1 = strain_avg + R\n", "ep2 = strain_avg - R\n", "tan_thetap =math.atan(-tou/(-ep_x + strain_avg))\n", "thetap = tan_thetap/2.0\n", "thetap2 = thetap*(180/math.pi)\n", "\n", "#Display\n", "print'The maximum in-plane principal strains are',round(ep1,0),\"*10**-6 and \",round(ep2,1),\"*10**-6\"\n", "print'The angle of orientation ',round(thetap2,1),\"degree\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The maximum in-plane principal strains are 272.0 *10**-6 and 33.8 *10**-6\n", "The angle of orientation 19.3 degree\n" ] } ], "prompt_number": 76 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.9 Page No 512" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "#Given\n", "E_st = 200*10**9 #GPa\n", "nu_st = 0.3 #Poisson's ratio\n", "ep1 = 272 *10**-6\n", "ep2 = 33.8 *10**-6\n", "\n", "#Solving for the equations\n", "#6.78*10**-6=sigma2-0.3sigma1\n", "#54.4*10**-6=sigma1-0.3sigma2\n", "sigma2= 25.4\n", "\n", "#Display\n", "print'The principal stresses at point A are ',sigma1,\"MPa and \" ,sigma2,\"Mpa\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The principal stresses at point A are 62 MPa and 25.4 Mpa\n" ] } ], "prompt_number": 79 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.10 Page No 514" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "a = 300.0 #mm\n", "b = 50.0 #mm\n", "t = 20.0 #mm\n", "E_cu = 120*10**3 #MPa\n", "nu_cu = 0.34 # Poisson's ratio\n", "#By inspection\n", "sigma_x = 800 #MPa\n", "sigma_y = -500.0 #MPa\n", "tou_xy = 0\n", "sigma_z = 0\n", "\n", "#calculation\n", "#By Hooke's Law\n", "ep_x = (sigma_x/E_cu) - (nu_cu/E_cu)*(sigma_y + sigma_z)\n", "ep_y = (sigma_y/E_cu) - (nu_cu/E_cu)*(sigma_x + sigma_z)\n", "ep_z = (sigma_z/E_cu) - (nu_cu/E_cu)*(sigma_y + sigma_x)\n", "#New lengths\n", "a_dash = a + ep_x*a\n", "b_dash = b + ep_y*b\n", "t_dash = t + ep_z*t\n", "\n", "#Display\n", "print'The new length = ',round(a_dash,1),\"mm\"\n", "print 'The new nas base is',round(b_dash,1),\"mm\"\n", "print'The new thickness = ',round(t_dash,2),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The new length = 300.4 mm\n", "The new nas base is 49.7 mm\n", "The new thickness = 19.98 mm\n" ] } ], "prompt_number": 88 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.11 Page No 515" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "p = 20 #psi, pressure\n", "E = 600 #psi, pressure\n", "nu = 0.45\n", "#the given dimension are:\n", "a = 4 #in\n", "b = 2 # in\n", "c = 3 #in\n", "\n", "#Calculation\n", "#Dilatation\n", "sigma_x = -p\n", "sigma_y = -p\n", "sigma_z = -p\n", "e = ((1-2*nu)/E)*(sigma_x + sigma_y + sigma_z)\n", "#Change in Length\n", "ep = (sigma_x - nu*(sigma_y + sigma_z))/E\n", "del_a = ep*a\n", "del_b = ep*b\n", "del_c = ep*c\n", "\n", "#Display\n", "print'The change in length a = ',round(del_a,4),\"inch\"\n", "print'The change in length b = ',round(del_b,4),\"inch\"\n", "print'The change in length c = ',round(del_c,4),\"inch\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The change in length a = -0.0133 inch\n", "The change in length b = -0.0067 inch\n", "The change in length c = -0.01 inch\n" ] } ], "prompt_number": 92 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.12 Page No 526" ] }, { "cell_type": "code", "collapsed": false, "input": [ "\n", "#Given\n", "T = 400 #lbft, tourqe\n", "sigma_ult = 20000 #psi\n", "\n", "#Calculations\n", "import math\n", "x = T*12/(math.pi/2)\n", "r=(x/sigma_ult)**(1/3.0)\n", "\n", "#Display\n", "print'The smallest radius of the solid cast iron shaft ',round(r,3),\"inch\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The smallest radius of the solid cast iron shaft 0.535 inch\n" ] } ], "prompt_number": 99 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "EXample 10.13 Page No 527" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Given\n", "import math\n", "sigmay=36 #ksi, stress\n", "r = 0.5 #cm\n", "sigma_yield = 360 #MPa, yield stress\n", "T = 3.25 #kN/cm\n", "A= (math.pi*r**2)\n", "P = 15 #kN\n", "J = (math.pi/2.0)*(r**4)\n", "sigma_y_sqr = sigma_yield**2\n", "\n", "#Calculations\n", "sigma_x = -(P/A)\n", "sigma_y = 0\n", "tou_xy = (T*r)/J\n", "k = (sigma_x + sigma_y)/2.0\n", "R = sqrt(k**2 + (tou_xy**2))\n", "sigma1 = k+R\n", "sigma2 = k-R\n", "l = sigma1 - sigma2\n", "#Maximum Shear Stress Theory\n", "x=sigma1-sigma2\n", "y=sigma1**2+sigma2**2-sigma1*sigma2\n", "if x>sigmay:\n", " print\"Shear failure of material will occur\"\n", "else:\n", " print\"not\"\n", "if y