{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 7 : Transformations of Stress and Strain" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 7.01, Page number 446" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Principal planes = 26.550000 degree\n", "Case(a) Principal planes = 116.550000 degree\n", "Case(b) Maximum principal stress = 70.000000 MPa\n", "Case(b) Minimum principal stress = -30.000000 MPa\n", "Case(c) Maximum shearing stress = 50.000000 MPa\n" ] } ], "source": [ "import math\n", "from sympy import symbols,cos,sin\n", "\n", "#Variable declaration\n", "Sx=50 # Stress in x(MPa)\n", "n=-1\n", "Sy=n*10 # Stress in y(MPa)\n", "txy=40 # Shearing stress(MPa) \n", "\n", "\n", "#Calculation \n", "# Case(a) Principal axes\n", "phy1=(53.1)/2.0 # Angle(degree) \n", "phy2=(233.1)/2.0 # Angle(degree)\n", "\n", "# Case(b) Principal stresses\n", "Smax=20+math.sqrt(30**2+40**2) # Maximum principal stress(MPa)\n", "Smin=20-math.sqrt(30**2+40**2) # Maximum principal stress(MPa)\n", "Sxl=(50-10)/(2.0) + ((50+10)/(2.0))*cos(53.1*((math.pi)*2)/(360.0)) +40*sin(53.1*((math.pi)*2)/(360.0)) \n", "\n", "# Case(c) Maximum shearing stress\n", "tmax=math.sqrt(30**2+40**2) # Maximum shearing stress(MPa)\n", "Sl=(50-10)/2.0 # Normal stress on each of four walls(MPa)\n", "\n", "# Result\n", "print ('Case(a) Principal planes = %lf degree' %phy1)\n", "print ('Case(a) Principal planes = %lf degree' %phy2)\n", "print ('Case(b) Maximum principal stress = %lf MPa' %Smax)\n", "print ('Case(b) Minimum principal stress = %lf MPa' %Smin)\n", "print ('Case(c) Maximum shearing stress = %lf MPa' %tmax)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 7.1, Page number 447" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Normal stress = 8.840000 ksi\n", "Case(a) Shearing stress = 7.960000 ksi\n", "Case(b) Principal plane angle = -30.500000 degree\n", "Case(b) Principal plane angle = 119.000000 degree\n", "Case(c) Maximum stress at point H = 13.524834 ksi\n", "Case(c) Minimum stress at point H = -4.684834 ksi\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "P=150 # Horizontal force(lb)\n", "T=(150*18)/(1000.0) # Force couple system(kip.in)\n", "Mx=(150*10)/(1000.0) # Force couple system(kip.in)\n", "sx=0 # Stress at x\n", "M=1.5 # Torque(kip.in)\n", "c=0.6 # Radius(in)\n", "n=-1\n", "\n", "#Calculation \n", "#Case(a) Stresses S x , S y , T xy at Point H\n", "sy=round(((M)*(c))/((1/4.0)*(math.pi)*(pow(0.6,4))),2) # Stress(ksi)\n", "txy=round(((T)*(c))/((1/2.0)*(math.pi)*(pow(0.6,4))),2) # Shearing stress(ksi)\n", "\n", "#Case(b) Principal Planes and Principal Stresses\n", "phyp1=(n*61)/2.0 # Angle(degree)\n", "phyp2=180-61 # Angle(degree)\n", "\n", "Smax=8.84/2.0 + math.sqrt(4.42**2 + 7.96**2) # Maximum stress(ksi) \n", "Smin=8.84/2.0 - math.sqrt(4.42**2 + 7.96**2) # Minimum stress(ksi)\n", "\n", "# Result\n", "print ('Case(a) Normal stress = %lf ksi' %sy)\n", "print ('Case(a) Shearing stress = %lf ksi' %txy)\n", "print ('Case(b) Principal plane angle = %lf degree' %phyp1)\n", "print ('Case(b) Principal plane angle = %lf degree' %phyp2)\n", "print ('Case(c) Maximum stress at point H = %lf ksi' %Smax)\n", "print ('Case(c) Minimum stress at point H = %lf ksi' %Smin)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 7.02, Page number 454" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) radius of circle = 50.000000 MPa\n", "Case(b) Maximum principal stress = 70.000000 MPa\n", "Case(b) Minimum principal stress = -30.000000 MPa\n", "Case(c) Maximum shearing stress = 50.000000 MPa\n", "Case(c) Minimum shearing stress = 20.000000 MPa\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "#Variable declaration\n", "n=-1\n", "Sx=50 # Force at x(MPa)\n", "Sy=n*10 # Force at y(MPa)\n", "OC=20 # Force(MPa) \n", "CA=50 # Force(MPa)\n", "BC=50 # Force(MPa)\n", "\n", "#Calculation \n", "# Case(a) Construction of Mohr’s Circle\n", "Save=(Sx+Sy)/2.0 # Average stress(MPa)\n", "CF=50-20 # Force(MPa) \n", "FX=40 # Force(MPa) \n", "R=math.sqrt(30**2 + 40**2) # Radius of circle(MPa)\n", "# case(b) Principal Planes and Principal Stresses\n", "Smax=OC+CA # Maximum stress(MPa) \n", "Smin=OC-BC # Minimum stress(MPa)\n", "phyp=(53.1)/2.0 # Angle(degree)\n", "# case(c) Maximum Shearing Stress\n", "tmax=50 # Maximum shearing stress(MPa) \n", "Save=20 # Maximum normal stress(MPa) \n", "\n", "# Result\n", "print ('Case(a) radius of circle = %lf MPa' %R)\n", "print ('Case(b) Maximum principal stress = %lf MPa' %Smax)\n", "print ('Case(b) Minimum principal stress = %lf MPa' %Smin)\n", "print ('Case(c) Maximum shearing stress = %lf MPa' %tmax)\n", "print ('Case(c) Minimum shearing stress = %lf MPa' %Save)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 7.2, Page number 457" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Principal planes angle = 33.700000 MPa\n", "Case(b) Maximum principal stress = 132.000000 MPa\n", "Case(b) Minimum principal stress = 28.000000 MPa\n", "Case(c) Stress in x direction = 48.416456 MPa\n", "Case(c) Stress in y direction = 111.583544 MPa\n", "Case(c) Stress in x and y direction = 41.309560 MPa\n" ] } ], "source": [ "import math\n", "from sympy import symbols,cos,sin\n", "\n", "#Variable declaration\n", "sx=100 # Force(MPa)\n", "sy=60 # Force(MPa)\n", "CF=20 # Force(MPa)\n", "FX=48 # Force(MPa) \n", "OC=80 # Force(MPa)\n", "CA=52 # Force(MPa)\n", "BC=52 # Force(MPa) \n", "\n", "# Calculation\n", "#Construction of Mohr’s Circle\n", "R=math.sqrt(20**2+48**2) # Radius of circle(MPa)\n", "\n", " \n", "#Case(a) Principal Planes and Principal Stresses\n", "phyp=(67.4)/2.0 # Angle(degree)\n", "Smax=OC+CA # Maximum stress(MPa)\n", "Smin=OC-BC # Min stress(MPa)\n", "\n", "#Case(b) Stress Components on Element Rotated 30\n", "phy=180-60 # Angle(degree)\n", "Sxl=80-(52*(cos(52.6*(math.pi*2)/(360.0))))\n", "Syl=80+(52*(cos(52.6*(math.pi*2)/(360.0))))\n", "txlyl=52*(sin(52.6*(math.pi*2)/(360.0)))\n", "\n", "# Result\n", "print ('Case(a) Principal planes angle = %lf MPa' %phyp)\n", "print ('Case(b) Maximum principal stress = %lf MPa' %Smax)\n", "print ('Case(b) Minimum principal stress = %lf MPa' %Smin)\n", "print ('Case(c) Stress in x direction = %lf MPa' %Sxl)\n", "print ('Case(c) Stress in y direction = %lf MPa' %Syl)\n", "print ('Case(c) Stress in x and y direction = %lf MPa' %txlyl)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 7.3, Page number 458" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Shearing stress t0 for which the largest normal stress is 10 ksi = 4.472856 ksi\n", "Case(b) Maximum shearing stress = 6.000000 ksi\n" ] } ], "source": [ "import math\n", "from sympy import symbols,sin\n", "\n", "#Variable declaration\n", "S0=8 # Stress(ksi)\n", "Sx=8 # Stress in x direction(ksi)\n", "Sy=0 # Stress in y direction(ksi) \n", "Smax=10 # Maximum stress(ksi)\n", "\n", "# Calculation\n", "#Construction of Mohr’s Circle\n", "Save=(1/2.0)*(Sx+Sy) # Average stress(ksi) \n", "R=10-4 # Radius of mohr's circle(ksi) \n", "\n", "#Calculation \n", "#Case(a) Shearing Stress t0\n", "phyp=(48.2)/2.0 # Angle(degree) \n", "t0=R*sin(48.2*((math.pi)*2)/(360.0)) # Shearing stress(ksi)\n", "\n", "#Case(b) Maximum Shearing Stress\n", "tmax=R # Maximum shearing stress(ksi)\n", "phyx=(90-48.2)/(2.0) \n", "\n", "#Result\n", "print ('Case(a) Shearing stress t0 for which the largest normal stress is 10 ksi = %lf ksi' %t0)\n", "print ('Case(b) Maximum shearing stress = %lf ksi' %tmax)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 7.03, Page number 466" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Principal stress at A = 8.000000 ksi\n", "Case(a) Principal stress at B = 1.500000 ksi\n", "Case(b) Principal plane = 33.700000 ksi\n", "Case(c) Maximum shearing stress = 4.000000 ksi\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "n=-1\n", "Sx=6\n", "Sy=3.5\n", "OC=4.75\n", "CA=3.25\n", "BC=3.25\n", "\n", "#Calculation \n", "# Case(a) Principal Planes and Principal Stresses\n", "Save=(Sx+Sy)/2.0 # Average stress(ksi)\n", "R=math.sqrt(1.25**2 + 3**2) # Radius of circle(ksi)\n", "Sa=OC+CA # Principal stress(ksi)\n", "Sb=OC-BC # Principal stress(ksi) \n", "phyp=(67.4)/2.0\n", "\n", "# Case(b) Maximum shearing stress\n", "tmax=(1/2.0)*(Sa) # Maximum torque(ksi)\n", " \n", "#Result\n", "print ('Case(a) Principal stress at A = %lf ksi' %Sa)\n", "print ('Case(a) Principal stress at B = %lf ksi' %Sb)\n", "print ('Case(b) Principal plane = %lf ksi' %phyp)\n", "print ('Case(c) Maximum shearing stress = %lf ksi' %tmax)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 7.4, Page number 472" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Factor of safety with respect to maximum shearing stress criteria = 1.923077 \n", "Case(b) Factor of safety with respect to maximum-distortion-energy criterion = 2.190000 \n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "#Mohr’s Circle\n", "Sx=80 # Stress at x(MPa)\n", "sy=-40 # Stress at y(MPa) \n", "CF=60 # Stress(MPa) \n", "FX=25 # Stress(MPa)\n", "OC=20 # Stress(MPa)\n", "CA=65 # Stress(MPa)\n", "BC=65 # Stress(MPa)\n", "Save=(1/2.0)*(sx+sy) # Stress average(MPa) \n", "R=math.sqrt(60**2+25**2) # radius of mohr's circle(MPa) \n", "Sy=250 # Tensile strength(MPa) \n", "tm=65 # Stress(MPa) \n", "\n", "#Principal Stresses\n", "sa=OC+CA # Stress at A(MPa)\n", "sb=OC-BC # Stress at B(MPa)\n", "\n", "#Calculation \n", "#Case(a) Maximum-Shearing-Stress Criterion # \n", "ty=(1/2.0)*(Sy) # Shearing stress at yield(MPa)\n", "FS1=(ty/tm) # Factor of safety\n", "\n", "#Case(b) Maximum-Distortion-Energy Criterion\n", "FS2=round(math.sqrt((250**2)/(85.0**2+45.0**2+85.0*45)),2) # Factor of safety \n", "\n", "\n", "# Result\n", "print ('Case(a) Factor of safety with respect to maximum shearing stress criteria = %lf ' %FS1)\n", "print ('Case(b) Factor of safety with respect to maximum-distortion-energy criterion = %lf ' %FS2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 7.5, Page number 481" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Normal stress = 4230.000000 \n", "Case(a) Maximum shearing stress = 4230.000000 \n", "Case(b) Stress in direction perpendicular to helical weld = 4140.000000 \n", "Case(b) Stress in direction parallel to helical weld = 1344.000000 \n" ] } ], "source": [ "import math\n", "from sympy import symbols,cos,sin\n", "\n", "#Variable declaration\n", "p=180 # Internal gage pressure(psi)\n", "t=(5/16.0) # Length(in)\n", "r=(15-t) # Distance(in)\n", "\n", "\n", "\n", "#Calculation \n", "#Case(a) Spherical Cap\n", "s=((p)*(r))/(2.0*t) # Stress(psi)\n", "tmax=(1/2.0)*((p*r)/(t)) # Maximum shearing stress(psi) \n", "\n", "#Case(b) Cylindrical Body of the Tank\n", "t=3/8.0 # Distance(in) \n", "r=15-t # Distance(in) \n", "s1=(p*r)/(t) # Stress(psi)\n", "s2=(1/2.0)*s1 # Stress(psi)\n", "Save=(1/2.0)*(s1+s2) # Stress average(psi) \n", "R=(1/2.0)*(s1-s2) # Stress(psi) \n", "\n", "#Stresses at the Weld\n", "Sw=round(Save-(R*cos(50*(((math.pi)*2)/360.0))),-1) # Stress at the weld(psi)\n", "tw=round(R*sin(50*(((math.pi)*2)/360.0)),0) # Shearing stress at the weld(psi)\n", "\n", "# Result\n", "print ('Case(a) Normal stress = %lf ' %s)\n", "print ('Case(a) Maximum shearing stress = %lf ' %tmax)\n", "print ('Case(b) Stress in direction perpendicular to helical weld = %lf ' %Sw)\n", "print ('Case(b) Stress in direction parallel to helical weld = %lf ' %tw)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 7.04, Page number 490" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Principal strain = 483.000000 u\n", "Case(a) Principal strain = -83.000000 u\n", "Case(b) Maximum shearing strain = 566.000000 u \n", "Case(b) Normal strain = 200.000000 u\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "n=-1\n", "Sx=6 # Stress in x direction()\n", "Sy=3.5 # Stress in y direction()\n", "BC=3.25 \n", "\n", "#Calculation \n", "#Case(a) Principal Axes and Principal Strains\n", "Ex=4*(pow(10,-6)) # Distance(u) \n", "Ey=0 # Distance(u)\n", "OC=(Ex+Ey)/(2.0) # Radius(u) \n", "OY=200 # Radius(u)\n", "\n", "R=math.sqrt(200**2 + 200**2) # Radius(u)\n", "\n", "Ea=200 + 283 # Strain at point A(u)\n", "Eb=200 - 283 # Strain at point A(u)\n", "\n", "#Case(b) Maximum shearing strain\n", "ymax=283*2 # Maximum shearing strain(u)\n", "El=200 # Normal strain(u)\n", "\n", "# Result\n", "print ('Case(a) Principal strain = %lf u' %Ea)\n", "print ('Case(a) Principal strain = %lf u' %Eb)\n", "print ('Case(b) Maximum shearing strain = %lf u ' %ymax)\n", "print ('Case(b) Normal strain = %lf u' %El)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 7.05, Page number 494" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Maximum in plane shearing strain = 0.000450 rad\n", "Case(b) Maximum shearing strain = -0.000150 u\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "#Variable declaration\n", "Ea=400*(pow(10,-6)) # Principal strain(in./in)\n", "Eb=-50*(pow(10,-6)) # Principal strain(in./in)\n", "v=0.30 # Poisson's ratio\n", "n=-1\n", "\n", "#Calculation \n", "#Case(a) Maximum In-Plane Shearing Strain\n", "Ymaxinplane=Ea-Eb # Maximum in-plane shearing strain(rad)\n", "#Case(b) Maximum Shearing Strain\n", "Ec=n*(v/(1.0-v))*(Ea+Eb) # Strain(in./in) # Maximum shearing strain(rad)\n", "\n", "# Result\n", "print ('Case(a) Maximum in plane shearing strain = %lf rad' %Ymaxinplane)\n", "print ('Case(b) Maximum shearing strain = %lf u' %Ec)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 7.6, Page number 496" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Gage pressure inside the tank = 546.000000 psi\n", "Case(b) Principal stress = 4.370000 ksi\n", "Case(b) Principal stress = 8.740000 ksi\n", "Case(b) Maximum shearing stress = 4.370000 ksi\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "s1=symbols('s1') # Variable declaration\n", "s2=symbols('s2') # Variable declaration\n", "E1=255*(pow(10,-6)) # Strain(in./in) \n", "E2=60*(pow(10,-6)) # Strain(in./in)\n", "G=11.2*(pow(10,6)) # Modulus of rigidity(psi)\n", "\n", "#Calculation \n", "#Gage Pressure Inside Tank\n", "ymax=E1-E2 # Maximum in plane shearing strain(rad)\n", "tmax=(round(G*ymax,0))/(pow(10,3)) # Shearing stress(ksi) \n", "p=(2184*4*0.75)/12.0 # Gage pressure(psi)\n", "\n", "#Principal Stresses and Maximum Shearing Stress\n", "s2=round(2*tmax,2) # Principal Stress(ksi)\n", "s1=round(2*s2,2) # Principal Stress(ksi)\n", "tmax=(1/2.0)*(s1) # Shearing stress(ksi) \n", "\n", "# Result\n", "print ('Case(a) Gage pressure inside the tank = %lf psi' %p)\n", "print ('Case(b) Principal stress = %lf ksi' %s2)\n", "print ('Case(b) Principal stress = %lf ksi' %s1)\n", "print ('Case(b) Maximum shearing stress = %lf ksi' %tmax)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 7.7, Page number 497" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a) Strain component are :-\n", "{ey: 860.050456293436, ex: 40.0000000000000, yxy: 750.577367205543}\n", "Case(b) Principal strains = -106.000000 ksi\n", "Case(b) Principal strains = 1006.000000 ksi\n", "Case(b) Principal strains = -367.605634 ksi\n", "Case(c) Maximum shearing strain = 1374.000000 ksi\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "E1=40 # Strain(u) \n", "E2=980 # Strain(u) \n", "E3=330 # Strain(u)\n", "exp1=symbols('exp1') # Variable declaration\n", "exp2=symbols('exp2') # Variable declaration\n", "exp3=symbols('exp3') # Variable declaration\n", "ex=symbols('ex') # Variable declaration\n", "ey=symbols('ey') # Variable declaration\n", "yxy=symbols('yxy') # Variable declaration\n", "v=0.29\n", "n=-1\n", "\n", "#Calculation \n", "#Strain Components ex, ey, yxy\n", "exp1=ex*1 + ey*0 + yxy*0*1-40 # Strain equations\n", "exp2=ex*(pow(0.5,2)) + ey*(pow(0.866,2)) + yxy*(0.866*0.5)-980 # Strain equations\n", "exp3=ex*(pow(-0.5,2)) + ey*(pow(0.866,2)) + yxy*(0.866*-0.5)-330 # Strain equations\n", "Sol=solve((exp1,exp2,exp3),ex,ey,yxy) # \n", "\n", "\n", "#Principal Strains\n", "Eave=(1/2.0)*(860+40) # Strain(u)\n", "R=math.sqrt(375**2+410**2) # Radius(u)\n", "phyp=(42.4)/2.0 # Angle(degree)\n", "Ea=-106 # Strain(u)\n", "Eb=1006 # Strain(u)\n", "Ec=n*(v/(1-v))*(Ea+Eb) # Strain(u)\n", "\n", "#Maximum Shearing Strain\n", "Ymax=(1006+368)\n", "\n", "# Result\n", "print ('Case(a) Strain component are :-')\n", "print(Sol)\n", "print ('Case(b) Principal strains = %lf ksi' %Ea)\n", "print ('Case(b) Principal strains = %lf ksi' %Eb)\n", "print ('Case(b) Principal strains = %lf ksi' %Ec)\n", "print ('Case(c) Maximum shearing strain = %lf ksi' %Ymax)\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.1" } }, "nbformat": 4, "nbformat_minor": 0 }