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diff --git a/Mechanics_of_Materials/Chapter10.ipynb b/Mechanics_of_Materials/Chapter10.ipynb new file mode 100755 index 00000000..8492cf8c --- /dev/null +++ b/Mechanics_of_Materials/Chapter10.ipynb @@ -0,0 +1,674 @@ +{
+ "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<sigmay**2:\n",
+ " print\"Failure will not occur\"\n",
+ "else:\n",
+ " print\"it will occur\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Shear failure of material will occur\n",
+ "Failure will not occur\n"
+ ]
+ }
+ ],
+ "prompt_number": 14
+ }
+ ],
+ "metadata": {}
+ }
+ ]
+}
\ No newline at end of file |