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author | Trupti Kini | 2016-01-11 23:30:05 +0600 |
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committer | Trupti Kini | 2016-01-11 23:30:05 +0600 |
commit | 8fa3a985b0772b28f8e6e6f5aec18bca1c0f7332 (patch) | |
tree | 5a9269d094b4a98947094c7c23cee3dbb3ae6c68 /Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter8.ipynb | |
parent | e1dbc4e45afcfcd26275010a9c0008b5d7f7e5ea (diff) | |
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Added(A)/Deleted(D) following books
A Engineering_Mechanics_of_Solids_by_Popov_E_P/Chapter1.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter10.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter11.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter12.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter2.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter4.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter5.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter6.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter7.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter8.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter9.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/charpter_3.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/charpter_3_1.ipynb
A Engineering_Mechanics_of_Solids_by_Popov_E_P/screenshots/Untitled.png
A Engineering_Mechanics_of_Solids_by_Popov_E_P/screenshots/cap2.png
A Engineering_Mechanics_of_Solids_by_Popov_E_P/screenshots/cap3.png
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch1.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch10.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch11.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch12.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch2.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch3.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch4.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch5.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch6.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch7.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch8.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/Ch9.ipynb
A Microwave_Engineering_by_G._S._Raghuvanshi/screenshots/12refCoeff.png
A Microwave_Engineering_by_G._S._Raghuvanshi/screenshots/1RefCoeff.png
A Microwave_Engineering_by_G._S._Raghuvanshi/screenshots/6axialPhasVel.png
Diffstat (limited to 'Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter8.ipynb')
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diff --git a/Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter8.ipynb b/Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter8.ipynb new file mode 100644 index 00000000..fb95b611 --- /dev/null +++ b/Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter8.ipynb @@ -0,0 +1,332 @@ +{
+ "metadata": {
+ "name": "",
+ "signature": "sha256:bed4afe1f989fb55f213b8e274b88cbf3d61242768410ad03d4984a97542fc14"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+ {
+ "cells": [
+ {
+ "cell_type": "heading",
+ "level": 1,
+ "metadata": {},
+ "source": [
+ "Chapter 8:Transformation of stress and strain and Yield and Fracture criteria "
+ ]
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 8.1 page number 405 "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "o = 22.5 #degrees , The angle of infetisimal wedge \n",
+ "A = 1 #mm2 The area of the element \n",
+ "A_ab = 1*(math.cos(radians(o))) #mm2 - The area corresponds to AB\n",
+ "A_bc = 1*(math.sin(radians(o))) #mm2 - The area corresponds to BC\n",
+ "S_1 = 3 #MN The stresses applying on the element \n",
+ "S_2 = 2 #MN\n",
+ "S_3 = 2 #MN\n",
+ "S_4 = 1 #MN \n",
+ "F_1 = S_1*A_ab # The Forces obtained by multiplying stress by their areas \n",
+ "F_2 = S_2*A_ab\n",
+ "F_3 = S_3*A_bc\n",
+ "F_4 = S_4*A_bc\n",
+ "#sum of F_N = 0 equilibrim in normal direction \n",
+ "N = (F_1-F_3)*(math.cos(radians(o))) + (F_4 - F_2)*(math.sin(radians(o)))\n",
+ "\n",
+ "#sum of F_s = 0 equilibrim in tangential direction \n",
+ "\n",
+ "S = (F_2-F_4)*(math.cos(radians(o))) + (F_1 - F_3)*(math.sin(radians(o)))\n",
+ "\n",
+ "Stress_Normal = N/A #Mpa - The stress action in normal direction on AB\n",
+ "Stress_tan = S/A #Mpa - The stress action in tangential direction on AB\n",
+ "print \"The stress action in normal direction on AB\",round(Stress_Normal,2),\"Mpa\"\n",
+ "print \"The stress action in tangential direction on AB\",round(Stress_tan,2),\"Mpa\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The stress action in normal direction on AB 1.29 Mpa\n",
+ "The stress action in tangential direction on AB 2.12 Mpa\n"
+ ]
+ }
+ ],
+ "prompt_number": 6
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 8.2 page number 413"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given\n",
+ "o = -22.5 #degrees , The angle of infetisimal wedge \n",
+ "A = 1 #mm2 The area of the element \n",
+ "A_ab = 1*(math.cos(radians(o))) #mm2 - The area corresponds to AB\n",
+ "A_bc = 1*(math.sin(radians(o))) #mm2 - The area corresponds to BC\n",
+ "S_1 = 3.0 #MN The stresses applying on the element \n",
+ "S_2 = 2.0 #MN\n",
+ "S_3 = 2.0 #MN\n",
+ "S_4 = 1.0 #MN\n",
+ "#Caliculations \n",
+ "\n",
+ "F_1 = S_1*A_ab # The Forces obtained by multiplying stress by their areas \n",
+ "F_2 = S_2*A_ab\n",
+ "F_3 = S_3*A_bc\n",
+ "F_4 = S_4*A_bc\n",
+ "#sum of F_N = 0 equilibrim in normal direction \n",
+ "N = (F_1-F_3)*(math.cos(radians(o))) + (F_4 - F_2)*(math.sin(radians(o)))\n",
+ "\n",
+ "#sum of F_s = 0 equilibrim in tangential direction \n",
+ "\n",
+ "S = (F_2-F_4)*(math.cos(radians(o))) + (F_1 - F_3)*(math.sin(radians(o)))\n",
+ "\n",
+ "Stress_Normal = N/A #Mpa - The stress action in normal direction on AB\n",
+ "Stress_tan = S/A #Mpa - The stress action in tangential direction on AB\n",
+ "print \"a) The stress action in normal direction on AB\",round(Stress_Normal,2),\"Mpa\"\n",
+ "print \"a) The stress action in tangential direction on AB\",round(Stress_tan,2),\"Mpa\"\n",
+ "\n",
+ "#Part- b\n",
+ "\n",
+ "S_max = (S_4+S_1)/2 + (((((S_4-S_1)/2)**2) + S_3**2)**0.5) #Mpa - The maximum stress\n",
+ "S_min = (S_4+S_1)/2.0 - (((((S_4-S_1/2))**2) + S_3**2)**0.5) #Mpa - The minumum stress\n",
+ "k = 0.5*math.atan(S_3/((S_1-S_4)/2)) #radians The angle of principle axis\n",
+ "k_1 = math.degrees(k)\n",
+ "k_2 = k_1+90 #The principle plane angles\n",
+ "print \"b) The principle stress \",round(S_max,1),\"Mpa tension\"\n",
+ "print \"b) The principle stress \",round(S_min,2),\"Mpa compression\"\n",
+ "print \"b) The principle plane angles are\",round(k_1,0),\",\",round(k_2,0),\"degrees\"\n",
+ "\n",
+ "#part-c\n",
+ "#The maximum shear stress case\n",
+ "t_xy = (((((S_4-S_1)/2)**2) + S_3**2)**0.5) #Mpa - The maximum shear stress case\n",
+ "K = 0.5*math.atan((-(S_1-S_4)/(2*S_3))) #radians The angle of principle axis\n",
+ "K_0 = math.degrees(K)\n",
+ "if K_0<0:\n",
+ " K_1 = K_0+90\n",
+ "else:\n",
+ " K_1 = K_0\n",
+ "K_2 = K_1+90 #PRinciple plain angles\n",
+ "T_xy = -((S_1-S_4)/2)*(math.sin(radians(2*K_1))) + ((S_4+S_1)/2)*(math.cos(radians(2*K_1))) # Shear stress\n",
+ "print \"c) The maximum shear is \",round(T_xy,2),\"Mpa\" \n",
+ "S_mat_a = array([round(S_max,1),round(S_min,1),0]) #MPa maximum stress matrix\n",
+ "S_mat_b = array([(S_4+S_1)/2,round(T_xy,2),round(T_xy,2),(S_4+S_1)/2]) #MPa maximum stress matrix at maximum shear\n",
+ "print \"a)\",S_mat_a,\"Mpa\"\n",
+ "print \"b)\",S_mat_b,\"Mpa\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "a) The stress action in normal direction on AB 4.12 Mpa\n",
+ "a) The stress action in tangential direction on AB 0.71 Mpa\n",
+ "b) The maximum stress 4.2 Mpa tension\n",
+ "b) The minumum stress -0.06 Mpa compression\n",
+ "b) The principle plane angles are 32.0 , 122.0 degrees\n",
+ "c) The maximum shear is -2.24 Mpa\n",
+ "a) [ 4.2 -0.1 0. ] Mpa\n",
+ "b) [ 2. -2.24 -2.24 2. ] Mpa\n"
+ ]
+ }
+ ],
+ "prompt_number": 58
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 8.3 page number 421"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "S_x = -2 #Mpa _ the noraml stress in x direction\n",
+ "S_y = 4 #Mpa _ the noraml stress in Y direction\n",
+ "c = (S_x + S_y)/2 #Mpa - The centre of the mohr circle \n",
+ "point_x = -2 #The x coordinate of a point on mohr circle\n",
+ "point_y = 4 #The y coordinate of a point on mohr circle\n",
+ "Radius = pow((point_x-c)**2 + point_y**2,0.5) # The radius of the mohr circle\n",
+ "S_1 = Radius +1#MPa The principle stress\n",
+ "S_2 = -Radius +1 #Mpa The principle stress\n",
+ "S_xy_max = Radius #Mpa The maximum shear stress\n",
+ "print \"The principle stresses are\",S_1 ,\"Mpa\",S_2,\"Mpa\"\n",
+ "print \"The maximum shear stress\",S_xy_max,\"Mpa\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The principle stresses are 6.0 Mpa -4.0 Mpa\n",
+ "The maximum shear stress 5.0 Mpa\n"
+ ]
+ }
+ ],
+ "prompt_number": 61
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 8.4 page number 423"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given\n",
+ "import math \n",
+ "S_x = 3.0 #Mpa _ the noraml stress in x direction\n",
+ "S_y = 1.0 #Mpa _ the noraml stress in Y direction\n",
+ "c = (S_x + S_y)/2 #Mpa - The centre of the mohr circle \n",
+ "point_x = 1 #The x coordinate of a point on mohr circle\n",
+ "point_y = 3 #The y coordinate of a point on mohr circle\n",
+ "#Caliculations \n",
+ "\n",
+ "Radius = pow((point_x-c)**2 + point_y**2,0.5) # The radius of the mohr circle\n",
+ "#22.5 degrees line is drawn \n",
+ "o = 22.5 #degrees \n",
+ "a = 71.5 - 2*o #Degrees, from diagram \n",
+ "stress_n = c + Radius*math.sin(math.degrees(o)) #Mpa The normal stress on the plane \n",
+ "stress_t = Radius*math.cos(math.degrees(o)) #Mpa The tangential stress on the plane\n",
+ "print \"The normal stress on the 221/2 plane \",round(stress_n,2),\"Mpa\"\n",
+ "print \"The tangential stress on the 221/2 plane \",round(stress_t,2),\"Mpa\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The normal stress on the 221/2 plane 4.82 Mpa\n",
+ "The tangential stress on the 221/2 plane 1.43 Mpa\n"
+ ]
+ }
+ ],
+ "prompt_number": 84
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 8.7 page number 437"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "e_x = -500 #10-6 m/m The contraction in X direction\n",
+ "e_y = 300 #10-6 m/m The contraction in Y direction\n",
+ "e_xy = -600 #10-6 m/m discorted angle\n",
+ "centre = (e_x + e_y)/2 #10-6 m/m \n",
+ "point_x = -500 #The x coordinate of a point on mohr circle\n",
+ "point_y = 300 #The y coordinate of a point on mohr circle\n",
+ "Radius = 500 #10-6 m/m - from mohr circle\n",
+ "e_1 = Radius +centre #MPa The principle strain\n",
+ "e_2 = -Radius +centre #Mpa The principle strain\n",
+ "k = math.atan(300.0/900) # from geometry\n",
+ "k_1 = math.degrees(k)\n",
+ "print \"The principle strains are\",e_1,\"um/m\",e_2,\"um/m\"\n",
+ "print \"The angle of principle plane\",round(k_1,2) ,\"degrees\"\n",
+ "\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The principle strains are 400 um/m -600 um/m\n",
+ "The angle of principle plane 18.43 degrees\n"
+ ]
+ }
+ ],
+ "prompt_number": 87
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 8.8 page number 441"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "#Given\n",
+ "e_0 = -500 #10-6 m/m \n",
+ "e_45 = 200 #10-6 m/m \n",
+ "e_90 = 300 #10-6 m/m\n",
+ "E = 200 #Gpa - youngs modulus of steel \n",
+ "v = 0.3 # poissions ratio \n",
+ "#Caliculations \n",
+ "\n",
+ "e_xy = 2*e_45 - (e_0 +e_90 ) #10-6 m/m from equation 8-40 in text\n",
+ "# from example 8.7\n",
+ "e_x = -500 #10-6 m/m The contraction in X direction\n",
+ "e_y = 300 #10-6 m/m The contraction in Y direction\n",
+ "e_xy = -600 #10-6 m/m discorted angle\n",
+ "centre = (e_x + e_y)/2 #10-6 m/m \n",
+ "point_x = -500 #The x coordinate of a point on mohr circle\n",
+ "point_y = 300 #The y coordinate of a point on mohr circle\n",
+ "Radius = 500 #10-6 m/m - from mohr circle\n",
+ "e_1 = Radius +centre #MPa The principle strain\n",
+ "e_2 = -Radius +centre #Mpa The principle strain\n",
+ "\n",
+ "stress_1 = E*(10**-3)*(e_1+v*e_2)/(1-v**2) #Mpa the stress in this direction \n",
+ "stress_2 = E*(10**-3)*(e_2+v*e_1)/(1-v**2) #Mpa the stress in this direction \n",
+ "print\"The principle stresses are \",round(stress_1,2),\"Mpa\",round(stress_2,2),\"MPa\" "
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The principle stresses are 48.35 Mpa -105.49 MPa\n"
+ ]
+ }
+ ],
+ "prompt_number": 91
+ }
+ ],
+ "metadata": {}
+ }
+ ]
+}
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