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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 deleted file mode 100755 index fb95b611..00000000 --- a/Engineering_Mechanics_of_Solids_by_Popov_E_P/chapter8.ipynb +++ /dev/null @@ -1,332 +0,0 @@ -{
- "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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