{
"metadata": {
"name": ""
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 9:Stress Transformation"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.1 Page no 440"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"import math\n",
"tou = 25\t\t #MPa, shear stress\n",
"sigma1 = 50 \t\t#MPa, stress\n",
"sigma2 = 80 \t\t#MPa\n",
"phi = 30*(math.pi/180.0)\n",
"\n",
"# Calculations\n",
"sigma_x1 = (sigma1*math.cos(phi)*math.cos(phi))- (tou*math.cos(phi)*math.sin(phi)) - (sigma2*math.sin(phi)*math.sin(phi))- (tou*math.sin(phi)*math.cos(phi))\n",
"tou1 = (sigma1*math.cos(phi)*math.sin(phi))+ (tou*math.cos(phi)*math.cos(phi)) + (sigma2*math.sin(phi)*math.cos(phi))- (tou*math.sin(phi)*math.sin(phi))\n",
"sigma_x2 = (tou*math.cos(phi)*math.sin(phi))- (sigma2*math.cos(phi)*math.cos(phi)) + (tou*math.sin(phi)*math.cos(phi))+ (sigma1*math.sin(phi)*math.sin(phi))\n",
"tou2 = (tou*math.cos(phi)*math.cos(phi))+ (sigma2*math.cos(phi)*math.sin(phi)) - (tou*math.sin(phi)*math.sin(phi))+ (sigma1*math.sin(phi)*math.cos(phi))\n",
"\n",
"#Display\n",
"print\"The normal stress component in the x diection is = \",round(sigma_x1,1),\"MPa\"\n",
"print\" The shear stress component in the x diection is = \",round(tou1,1),\"MPa\"\n",
"print\" The normal stress component in the y diection is = \",round(sigma_x2,1),\"MPa\"\n",
"print\" The shear stress component in the y diection is = \",round(tou2,1),\"MPa\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The normal stress component in the x diection is = -4.2 MPa\n",
" The shear stress component in the x diection is = 68.8 MPa\n",
" The normal stress component in the y diection is = -25.8 MPa\n",
" The shear stress component in the y diection is = 68.8 MPa\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.2 Page no 444"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"import math\n",
"phi = -30*(math.pi/180) #angle\n",
"theta = 60*(math.pi/180) \n",
"sigma_x = -80 #MPa\n",
"sigma_y = 50 #MPa\n",
"tou_xy = -25 #MPa\n",
"\n",
"#Plane CD\n",
"sigma_x1 = (sigma_x+sigma_y)/2 + ((sigma_x-sigma_y)*math.cos(2*phi))/2 + (tou_xy*math.sin(2*phi)) #Eqn 9.1\n",
"tou_xy1 = ((-(sigma_x - sigma_y)*math.sin(2*phi))/2) + (tou_xy*math.cos(2*phi)) #Eqn 9.2\n",
"\n",
"#Plane BC\n",
"sigma_x2 = (sigma_x+sigma_y)/2 + ((sigma_x-sigma_y)*math.cos(2*theta))/2 + (tou_xy*math.sin(2*theta)) #Eqn 9.1\n",
"tou_xy2 = (-(sigma_x - sigma_y)*math.sin(2*theta))/2 + tou_xy*math.cos(2*theta) #Eqn 9.2\n",
"\n",
"#Display\n",
"print'The normal stress of plane CD inclined at 30 degrees = ',round(sigma_x1,1),\"MPa\"\n",
"print'The shear stress of plane CD inclined at 30 degrees = ',round(tou_xy1,1),\"MPa\"\n",
"print'The normal stress of plane BC inclined at 60 degrees = ',round(sigma_x2,1),\"MPa\"\n",
"print'The shear stress of plane BC inclined at 60 degrees = ',round(tou_xy2,1),\"MPa\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The normal stress of plane CD inclined at 30 degrees = -25.8 MPa\n",
"The shear stress of plane CD inclined at 30 degrees = -68.8 MPa\n",
"The normal stress of plane BC inclined at 60 degrees = -4.2 MPa\n",
"The shear stress of plane BC inclined at 60 degrees = 68.8 MPa\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.3 Page no 448"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"sigma_x = -20 #MPa, stress\n",
"sigma_y = 90 #MPa\n",
"tou_xy = 60 #MPa\n",
"\n",
"#Orientation of Element\n",
"import math\n",
"theta_p2 = math.atan((2*tou_xy)/(sigma_x - sigma_y))\n",
"theta_p2 = theta_p2/2.0\n",
"theta_p1 = (180+2*theta_p2)/2.0\n",
"\n",
"#Principal Stresses\n",
"\n",
"sigma1 = ((sigma_x+sigma_y)/2.0)+(math.sqrt(((sigma_x - sigma_y)/2.0)**2 + tou_xy**2))\n",
"sigma2 = ((sigma_x+sigma_y)/2.0)- math.sqrt(((sigma_x-sigma_y)/2.0)**2 + tou_xy**2)\n",
"sigma_x2 = ((sigma_x+sigma_y)/2.0)+ (((sigma_x-sigma_y)/2.0)*math.cos(2*theta_p2)) + (tou_xy*math.sin(2*theta_p2))\n",
"\n",
"#Display\n",
"print\"The first principal stress is = \",round(sigma1,1),\"MPa\"\n",
"print\"The second principal stress is = \",round(sigma2,1),\"MPa\"\n",
"print'The normal stress acting on the 23.7 degrees plane = ',round(sigma_x2,1),\"MPa\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The first principal stress is = 116.4 MPa\n",
"The second principal stress is = -46.4 MPa\n",
"The normal stress acting on the 23.7 degrees plane = -43.3 MPa\n"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.4 Page no 449"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"sigma_x = -20.0 #MPa, stress along x\n",
"sigma_y = 90.0 #MPa stress along y\n",
"tou_xy =60.0 #Mpa, shear stress\n",
"\n",
"#Calculation\n",
"#Orientation of Element\n",
"import math\n",
"theta_s2 = math.atan(-(sigma_x - sigma_y)/(2*tou_xy))\n",
"theta_s2 = theta_s2/2.0\n",
"theta_s1 = math.pi + 2*theta_s2\n",
"theta_s1 = theta_s1/2.0\n",
"\n",
"#Maximum in plane Shear Stress\n",
"tou_max = (math.sqrt(((sigma_x - sigma_y)/2.0)**2 + tou_xy**2))\n",
"tou_xy1 = -(sigma_x - sigma_y)*(math.sin(2*theta_s2))/2.0 + (tou_xy*math.cos(2*theta_s2))\n",
"#Average Normal Stress\n",
"sigma_avg = (sigma_x+sigma_y)/2\n",
"\n",
"#Display\n",
"print\"The maximum in-plane shear stress is = \",round(tou_xy1,1),\"MPa\"\n",
"print\"The average normal stress is = \",round(sigma_avg,0),\"MPa\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The maximum in-plane shear stress is = 81.4 MPa\n",
"The average normal stress is = 35.0 MPa\n"
]
}
],
"prompt_number": 26
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.7 Page no 465"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"sigma_x = -12 #ksi, stress along x\n",
"sigma_y = 0\n",
"tou_xy = -6 #ksi, stress along xy\n",
"\n",
"#Calculation\n",
"#Construction of the circle\n",
"import math\n",
"sigma_avg = (sigma_x+sigma_y)/2.0\n",
"R = sqrt((-sigma_x+sigma_avg)**2 + (tou_xy)**2)\n",
"#Principal Stresses\n",
"sigma2 = -R+sigma_avg\n",
"sigma1 = R+sigma_avg\n",
"theta_p2 = math.atan((-tou_xy)/(-sigma_x+sigma_avg))\n",
"theta_p2 = theta_p2/2*(180/math.pi)\n",
"\n",
"#Display\n",
"print'The first principal stress is = ',round(sigma1,2),\"ksi\"\n",
"print'The second principal stress is = ',round(sigma2,2),\"ksi\"\n",
"print'The direction of the principal plane is = ',theta_p2,\"degree\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The first principal stress is = 2.49 ksi\n",
"The second principal stress is = -14.49 ksi\n",
"The direction of the principal plane is = 22.5 degree\n"
]
}
],
"prompt_number": 31
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.8 Page no 466"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"sigma_x = -20.0 #MPa\n",
"sigma_y = 90.0 #MPa\n",
"tou_xy = 60.0 #MPa\n",
"\n",
"#Construction of the circle\n",
"import math\n",
"sigma_avg = (sigma_x+sigma_y)/2\n",
"R = math.sqrt(((sigma_x-sigma_avg))**2 + (tou_xy)**2)\n",
"#Maximum In plane Shear Stress\n",
"tou_max = R\n",
"theta_s1 = math.atan(-(sigma_x - sigma_avg)/(tou_xy))\n",
"theta_s1 = theta_s1/2.0*(180/math.pi)\n",
"\n",
"#Display\n",
"print'The maximum in-plane shear stresses are = ',round(tou_max,1),\"MPa\"\n",
"print'The second principal stress = ',sigma_avg,\"MPa\"\n",
"print'The orientation of the element is = ',round(theta_s1,1),\"degree\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The maximum in-plane shear stresses are = 81.4 MPa\n",
"The second principal stress = 35.0 MPa\n",
"The orientation of the element is = 21.3 degree\n"
]
}
],
"prompt_number": 38
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.9 Page no 467"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Calculate normal stress and shear stress \n",
"\n",
"#Given\n",
"sigma_x = -8.0 #MPa\n",
"sigma_y = 12.0 #MPa\n",
"tou_xy = -6.0 #Mpa\n",
"\n",
"#Construction of the circle\n",
"import math\n",
"sigma_avg = (sigma_x+sigma_y)/2.0\n",
"R = math.sqrt( 10**2 + tou_xy**2)\n",
"#Stresses on 30 degree element\n",
"phi = math.atan(6/10.0)\n",
"psi = (math.pi/3.0) - phi\n",
"#On face BD\n",
"sigma_x1 = 2 - (R*math.cos(psi))\n",
"tou_xy1 = (R*math.sin(psi))\n",
"#On face DE\n",
"sigma_x2 = 2 + (R*math.cos(psi))\n",
"tou_xy2 = -(R*math.sin(psi))\n",
"\n",
"#Display\n",
"print'The normal stress on plane BD inclined at 30 degrees is = ',round(sigma_x1,1),\"ksi\"\n",
"print'The normal stress on plane DE inclined at 60 degrees is = ',round(sigma_x2,1),\"ksi\"\n",
"print'The shear stress is = ',round(tou_xy1,1),\"ksi\"\n",
"print'The shear stress is = ',round(tou_xy2,1),\"ksi\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The normal stress on plane BD inclined at 30 degrees is = -8.2 ksi\n",
"The normal stress on plane DE inclined at 60 degrees is = 12.2 ksi\n",
"The shear stress is = 5.7 ksi\n",
"The shear stress is = -5.7 ksi\n"
]
}
],
"prompt_number": 43
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.10 Page no 476"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"sigma_max = 32 #MPa\n",
"sigma_min = 0 #MPa\n",
"sigma_int = 16 #MPa\n",
"\n",
"#Calculation\n",
"tou_max = (sigma_max - sigma_min)/2 \n",
"sigma_avg = (sigma_max + sigma_min)/2 \n",
"tou_in_plane = (sigma_max - sigma_int)/2\n",
"sigma_avg2 = sigma_avg + (tou_in_plane)\n",
"\n",
"#Display\n",
"print 'The normal shears tress is', sigma_avg,\"MPa\"\n",
"print'The maximum absolute shear stress = ',tou_max,\"MPa\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The normal shears tress is 16 MPa\n",
"The maximum absolute shear stress = 16 MPa\n"
]
}
],
"prompt_number": 49
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 9.11 Page no 477"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"tou = 40 #psi\n",
"sigma = -20 #psi\n",
"\n",
"#Calculation\n",
"#Principal Stresses\n",
"import math\n",
"sigma_avg = sigma/2\n",
"R = sqrt( (-sigma + sigma_avg)**2 + tou**2)\n",
"sigma_max = sigma_avg + R \n",
"sigma_min = sigma_avg - R \n",
"theta = math.atan(tou/(-sigma+sigma_avg))\n",
"theta = theta/2\n",
"#Absolute Maximum Shear Stress\n",
"tou_max = (sigma_max - sigma_min)/2\n",
"sigma_avg = (sigma_max + sigma_min)/2\n",
"\n",
"#Display\n",
"print'The prinicpal stresses at the point are ',round(sigma_max,2),\"psi and\",round(sigma_min,1),\"psi\"\n",
"print'The absolute maximum shear stress at the point ',round(tou_max,1),\"psi\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The prinicpal stresses at the point are 31.23 psi and -51.2 psi\n",
"The absolute maximum shear stress at the point 41.2 psi\n"
]
}
],
"prompt_number": 56
}
],
"metadata": {}
}
]
}