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{
"metadata": {
"name": "",
"signature": "sha256:7ce3350f5dcc3b0641adb55040e49ecdbd34a727a892086466c616c0ec4732d8"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter2-Analysis of Stress(Equlibrium) "
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex4-pg54"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"#find the new stress tensor tau\n",
"import numpy\n",
"from numpy import linalg\n",
"## initialization of variables\n",
"\n",
"tau=([[200, 100, 0],\n",
" [100, 0, 0],\n",
" [0 ,0, 500]]) ## some units\n",
"theta=60. ## degrees\n",
"##calculations\n",
"theta1=theta/57.3\n",
"a=([[math.cos(theta1), math.sin(theta1), 0],\n",
" [-math.sin(theta1), math.cos(theta1), 0],\n",
" [0, 0, 1]])\n",
"b=numpy.transpose(a)\n",
"tau_new=numpy.dot(a,tau)\n",
"tau_new1=numpy.dot(tau_new,b)\n",
"## Results\n",
"print('The new stress tensor is')\n",
"print tau_new1"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The new stress tensor is\n",
"[[ 136.62361289 -136.59689227 0. ]\n",
" [-136.59689227 63.37638711 0. ]\n",
" [ 0. 0. 500. ]]"
]
},
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex5-pg61"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"find the octahedral at this point\n",
"## initialization of variables\n",
"import math\n",
"sigma_1=100. ##kg*f/cm^2\n",
"sigma_2=100. ##kg*f/cm^2\n",
"sigma_3=-200. ##kg*f/cm^2\n",
"## calculations\n",
"tau_oct=1/3.*math.sqrt((sigma_1-sigma_2)**2+(sigma_2-sigma_3)**2+(sigma_3-sigma_1)**2)\n",
"## Results\n",
"print'%s %.2f %s '%('Octahedra shear stress at the point is=',tau_oct,' kgf/cm^2')\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Octahedra shear stress at the point is= 141.42 kgf/cm^2 \n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex7-pg61"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"#check whether the invariants of stress sensor\n",
"import numpy\n",
"from numpy import linalg\n",
"## initialization of variable\n",
"tau=numpy.matrix([[200, 100, 0],\n",
" [100, 0, 0],\n",
" [0, 0, 500]]) ## some units\n",
"theta=60. ## degrees\n",
"##calculations\n",
"theta=theta*math.pi/180.\n",
"a=numpy.matrix([[math.cos(theta), math.sin(theta), 0],\n",
" [-math.sin(theta), math.cos(theta), 0],\n",
" [0, 0, 1]])\n",
"b=numpy.transpose(a)\n",
"tau_new=numpy.dot(a,tau)\n",
"tau_new1=numpy.dot(tau_new,b)\n",
"\n",
"## stress invariants :old \n",
"I1=tau[0,0]+tau[1,1]+tau[2,2]\n",
"I2=tau[0,0]*tau[1,1]+tau[1,1]*tau[2,2]+tau[2,2]*tau[0,0]-(tau[0,1]**2+tau[1,2]**2+tau[2,0]**2)\n",
"I3=tau[0,0]*tau[1,1]*tau[2,2]+2*tau[0,1]*tau[1,2]*tau[2,0]-(tau[0,0]*tau[1,2]**2+tau[1,1]*tau[2,0]**2+tau[2,2]*tau[0,1]**2)\n",
"\n",
"## stress invariants :new\n",
"I11=tau_new1[0,0]+tau_new1[0,0]+tau_new1[1,1]\n",
"I22=tau_new1[0,0]*tau_new1[1,1]+tau_new1[1,1]*tau_new1[2,2]+tau_new1[1,1]*tau_new1[0,0]-[tau_new1[0,1]**2+tau_new1[1,2]**2+tau_new1[1,0]**2]\n",
"I33=tau_new1[0,0]*tau_new1[1,1]*tau_new1[2,2]+2*tau_new1[0,1]*tau_new1[1,2]*tau_new1[2,0]-[tau_new1[0,0]*tau_new1[1,2]**2+tau_new1[1,1]*tau_new1[2,0]**2+tau_new1[2,2]*tau_new1[0,1]**2]\n",
"\n",
"## Results\n",
"print'%s %.2f %s %.2f %s %.2f %s %.2f %s %.2f %s %.2f' %('The invariants of old stress tensor are I1=',I1,' I2=',I2,' I3=',I3,' \\n and that of the new stress tensor are I1=',I11,' I2=',I22,' I3=',I33)\n",
"\n",
"print('\\n Hence the same stress tensor invariants')\n",
"\n",
"\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The invariants of old stress tensor are I1= 700.00 I2= 90000.00 I3= -5000000.00 \n",
" and that of the new stress tensor are I1= 336.60 I2= 11698.73 I3= -5000000.00\n",
"\n",
" Hence the same stress tensor invariants\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex8-pg67"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"## initialization of variables\n",
"#find the value of sigma 1 and sigma2 at biaxial yeilding and unaxial\n",
"sigma_3=0. ## kgf/cm**2\n",
"tau_oct=1500. ## kgf/cm**2\n",
"n=2 ## given that sigma_1=n*sigma_2\n",
"## calculations\n",
"sigma_2=1500.*3./(math.sqrt(2*n**2-2*n+2)) ## ## kgf/cm**2\n",
"sigma_1=n*sigma_2 ## kgf/cm**2 \n",
"sigma_0=4500./math.sqrt(2.) ## kgf/cm**2\n",
"## Results\n",
"print'%s %.2f %s %.2f %s %.2f %s '%('The necessary stresses sigma_1, sigma_2 for biaxial yielding are \\n ',sigma_2,' kgf/cm^2' '',sigma_1,' kgf/cm^2' and 'for uniaxial yielding sigma_0 ',sigma_0,'kgf/cm^2.')\n",
" \n",
"\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The necessary stresses sigma_1, sigma_2 for biaxial yielding are \n",
" 1837.12 kgf/cm^2 3674.23 for uniaxial yielding sigma_0 3181.98 kgf/cm^2. \n"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex9-pg68"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"##initialization of variables\n",
"#find the magnitude and direction of principal stress for the a b c\n",
"## part (a)\n",
"tau_xx=300 ## kgf/cm**2\n",
"tau_yy=0 ## kgf/cm**2\n",
"tau_xy=600 ## kgf/cm**2\n",
"##calculations\n",
"sigma_1=(tau_xx+tau_yy)/2.+math.sqrt((1./2.*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"sigma_2=(tau_xx+tau_yy)/2-math.sqrt((1/2.*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"Beta=math.atan(2*tau_xy/(tau_xx-tau_yy))\n",
"Beta=Beta*180/math.pi\n",
"##Results\n",
"print'%s %.2f%s %.2f %s %.2f %s'%('\\n Part (a) \\n The magnitude of principal stresses are',sigma_1,''and '',sigma_2,'kgf/cm^2' and' \\n the direction is given by 2*beta=',Beta,'')\n",
"\n",
"\n",
"##part (b)\n",
"tau_xx=1000 ## kgf/cm**2\n",
"tau_yy=150 ## kgf/cm**2\n",
"tau_xy=450 ## kgf/cm**2\n",
"## calculations\n",
"sigma_1=(tau_xx+tau_yy)/2+math.sqrt((1/2*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"sigma_2=(tau_xx+tau_yy)/2-math.sqrt((1/2*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"Beta=math.atan(2*tau_xy/(tau_xx-tau_yy))\n",
"Beta1=Beta*180./math.pi\n",
"## Results\n",
"print'%s %.2f %s %.2f %s %.2f %s '%('\\n Part (b) \\n The magnitude of principal stresses are',sigma_1,''and '',sigma_2,'kgf/cm^2' and' \\n the direction is given by 2*beta=',Beta1,'')\n",
"\n",
"## part (c)\n",
"tau_xx=-850 ## kgf/cm**2\n",
"tau_yy=350 ## kgf/cm**2\n",
"tau_xy=700 ## kgf/cm**2\n",
"## calculations\n",
"sigma_1=(tau_xx+tau_yy)/2+math.sqrt((1/2*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"sigma_2=(tau_xx+tau_yy)/2-math.sqrt((1/2*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"Beta=math.atan(2*tau_xy/(tau_xx-tau_yy))\n",
"Beta=Beta*57.3\n",
"## Results\n",
"print'%s %.2f %s %.2f %s %.2f %s '%('\\n Part (c) \\n The magnitude of principal stresses are',sigma_1,''and '',sigma_2,'kgf/cm^2' and' \\n the direction is given by 2*beta=',-Beta,'')\n",
" \n",
"\n",
"## wrong answers were given in textbook for part (b) (c)\n",
"\n",
"\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
" Part (a) \n",
" The magnitude of principal stresses are 768.47 -468.47 \n",
" the direction is given by 2*beta= 75.96 \n",
"\n",
" Part (b) \n",
" The magnitude of principal stresses are 1025.00 125.00 \n",
" the direction is given by 2*beta= 45.00 \n",
"\n",
" Part (c) \n",
" The magnitude of principal stresses are 450.00 -950.00 \n",
" the direction is given by 2*beta= 63.44 \n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex10-pg70"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"# initialization of variables\n",
"#find the intensity of diagonal tension\n",
"tau_xx= -1 # kgf/cm^2\n",
"tau_yy= 0 # kgf/cm^2\n",
"tau_xy= 7 # kgf/cm^2\n",
"# calculations \n",
"sigma_1=(tau_xx+tau_yy)/2.+math.sqrt((1/2.*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"sigma_2=(tau_xx+tau_yy)/2.-math.sqrt((1/2.*(tau_xx-tau_yy))**2+tau_xy**2)\n",
"x=sigma_1 # positive one is tension\n",
"if(sigma_2>sigma_1):\n",
" x=sigma_2\n",
"\n",
"# Results\n",
"print'%s %.2f %s'%('The diagonal tension is ',x,' kgf/cm^2')\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The diagonal tension is 6.52 kgf/cm^2\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex11-pg70"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"# initialization of variables\n",
"#find the state of stress at the joint\n",
"d=2 # m\n",
"l=10 # m\n",
"t=1 # cm\n",
"p=15 # kgf/cm^2\n",
"pitch= 2*math.pi #m\n",
"##calculations\n",
"w=2*math.pi*d/2. # m\n",
"theta=math.atan(w/(2*math.pi))\n",
"sigma_z=p*d*100./(4.*t)\n",
"sigma_th=p*d*100./(2.*t)\n",
"sigma_th_new=(sigma_th+sigma_z)/2.+(sigma_th-sigma_z)/2.*math.cos(2*theta)\n",
"tau_thz=(sigma_z-sigma_th)*math.sin(2.*theta)/2\n",
"# results\n",
"print'%s %.2f %s %.2f %s '%('At the junction, the normal and shear stresses are',sigma_th_new,'' and '',-tau_thz,' kgf/cm^2 \\n respectively, and the rivets must be designed for this')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"At the junction, the normal and shear stresses are 1125.00 375.00 kgf/cm^2 \n",
" respectively, and the rivets must be designed for this \n"
]
}
],
"prompt_number": 12
}
],
"metadata": {}
}
]
}
|
