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{
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"name": "",
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"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": {}
}
]
}
|
