{ "metadata": { "name": "", "signature": "sha256:b494877451d53f8b0ca30d008c3144520923dfdb33c6562fbcdeab1b4ca2b7ce" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter08:Stresses due to Combined Loading" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Examlple 8.8.1, Page No:275" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "p=125 #Pressure in psi\n", "r=24 #Radius of the vessel in inches\n", "t=0.25 #Thickness of the vessel in inches\n", "E=29*10**6 #Modulus of Elasticity in psi\n", "v=0.28 #poisson ratio\n", "\n", "#Calcualtions\n", "#Part 1\n", "sigma_c=p*r*t**-1 #Circumferential Stress in psi\n", "sigma_l=sigma_c/2 #Longitudinat Stress in psi\n", "e_c=E**-1*(sigma_c-(v*sigma_l)) #Circumferential Strain using biaxial Hooke's Law \n", "delta_r=e_c*r #Change in the radius in inches\n", "\n", "#Part 2\n", "sigma=(p*r)*(2*t)**-1 #Stress in psi\n", "e=E**-1*(sigma-(v*sigma)) #Strain using biaxial Hooke's Law\n", "delta_R=e*r #Change inradius of end-cap in inches\n", "\n", "#Result\n", "print \"Part 1 Answers\"\n", "print \"Stresses are sigma_c=\",round(sigma_c),\"psi and sigma_l=\",round(sigma_l),\"psi\"\n", "print \"Change of radius of cylinder=\",round(delta_r,5),\"in\"\n", "print \"Part 2 Answers\"\n", "print \"Stresses are sigma=\",round(sigma),\"psi\"\n", "print \"Change in radius of end cap=\",round(delta_R,5),\"in\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Part 1 Answers\n", "Stresses are sigma_c= 12000.0 psi and sigma_l= 6000.0 psi\n", "Change of radius of cylinder= 0.00854 in\n", "Part 2 Answers\n", "Stresses are sigma= 6000.0 psi\n", "Change in radius of end cap= 0.00358 in\n" ] } ], "prompt_number": 5 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.2, Page No:280" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "#Variable Decleration\n", "P=40 #Force in kN\n", "b=0.050 #Width in m\n", "h=0.040 #Depth in m\n", "\n", "#Calculations\n", "#Part 1\n", "A=b*h #Area in m^2\n", "I=(b*h**3)*12**-1 #Moment of inertia in m^4\n", "c=h*0.5 #m\n", "sigma_max=(P*A**-1)+(P*c**2*I**-1) #Maximum stress in MPa\n", "sigma_min=(P*A**-1)-(P*c**2*I**-1) #Minimum stress in MPa\n", "\n", "#Result\n", "print \"The Maximum and Minimum Stress are\"\n", "print \"Max=\",sigma_max/1000,\"MPa and Min=\",sigma_min/1000,\"MPa\"\n", "\n", "#Plotting\n", "x=[20,0,-20]\n", "S=[-sigma_min/1000,0,sigma_max/1000]\n", "plt.plot(S,x)\n", "plt.ylabel(\"Distance from Neutral Axis in mm\")\n", "plt.xlabel(\"Stress in MPa\")\n", "plt.title(\"Stress Distribution Diagram\")\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The Maximum and Minimum Stress are\n", "Max= 80.0 MPa and Min= -40.0 MPa\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 21 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.3, Page No:281" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variabel Decleration\n", "b=6 #Width in inches\n", "h=10 #Depth in inches\n", "P1=6000 #Force in lb\n", "P2=3000 #Force in lb\n", "L=4 #Length in ft\n", "P=-13400 #Load in lb\n", "M=6000 #Moment in lb.ft\n", "y=5 #Depth in inches\n", "P2=-9800 #Load in lb\n", "M2=-12000 #Moment in lb.ft\n", "\n", "#Calculations\n", "A=b*h #Area in in^2\n", "I=b*h**3*12**-1 #Moment of inertia in in^4\n", "T=(P1*L+P2*L*3)*(6)**-1 #Tension in the cable in lb\n", "\n", "#Computation of largest stress\n", "sigma_B=(P*A**-1)-(M*y*12*I**-1) #Maximum Compressive Stress caused by +ve BM in psi\n", "sigma_C=(P2*A**-1)-(M2*-y*12*I**-1) #Maximum Compressive Stress caused by -ve BM in psi\n", "\n", "sigma_max=max(-sigma_B,-sigma_C) #Maximum Compressive Stress in psi\n", "\n", "#Result\n", "print \"The maximum Stress is\",round(sigma_max),\"psi\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The maximum Stress is 1603.0 psi\n" ] } ], "prompt_number": 27 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.4, Page No:297" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "theta=(60*pi)/180 #Angle in radians (Twice as declared)\n", "sigma_x=30 # Stress in x in MPa\n", "sigma_y=60 #Stress in y in MPa\n", "tau_xy=40 #Stress in MPa\n", "\n", "#Calcualtions\n", "sigma_xdash=0.5*(sigma_x+sigma_y)+0.5*(sigma_x-sigma_y)*cos(theta)+tau_xy*sin(theta) #Stress at x' axis in MPa\n", "sigma_ydash=0.5*(sigma_x+sigma_y)-0.5*(sigma_x-sigma_y)*cos(theta)-tau_xy*sin(theta) #Stress at y' axis in MPa\n", "tau_x_y=-0.5*(sigma_x-sigma_y)*sin(theta)+tau_xy*cos(theta) #Stress at x'y' in shear in MPa\n", "#Result\n", "print \"The new stresses at new axes are as follows\"\n", "print \"sigma_x'=\",round(sigma_xdash,1),\"MPa sigma_y'=\",round(sigma_ydash,1),\"MPa\"\n", "print \"And tau_x'y'=\",round(tau_x_y),\"MPa\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The new stresses at new axes are as follows\n", "sigma_x'= 72.1 MPa sigma_y'= 17.9 MPa\n", "And tau_x'y'= 33.0 MPa\n" ] } ], "prompt_number": 22 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.5, Page No:297" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "sigma_x=8000 #Stress in x in psi\n", "sigma_y=4000 #Stress in y in psi\n", "tau_xy=3000 #Stress in xy in psi\n", "\n", "#Calculations\n", "R=sqrt(((sigma_x-sigma_y)*0.5)**2+tau_xy**2) #Resultant Stress in psi\n", "\n", "#Principal Stresses\n", "sigma1=(sigma_x+sigma_y)*0.5+R #Principal Stress in psi\n", "sigma2=(sigma_x+sigma_y)*0.5-R #Principal Stress in psi\n", "\n", "#Principal Direction\n", "theta1=arctan(2*tau_xy*(sigma_x-sigma_y)**-1)*0.5*180*pi**-1 #Principal direction in degrees\n", "theta2=theta1+90 #Second pricnipal direction in degrees\n", "\n", "#Normal Stress\n", "sigma_xdash=0.5*(sigma_x+sigma_y)+0.5*(sigma_x-sigma_y)*cos(2*theta1*pi*180**-1)+tau_xy*sin(2*theta1*pi*180**-1)\n", "\n", "#Result\n", "print \"The principal stresses are as follows\"\n", "print \"sigma1=\",round(sigma1),\"psi and sigma2=\",round(sigma2),\"psi\"\n", "print \"The corresponding directions are\"\n", "print \"Theta1=\",round(theta1,1),\"degrees and Theta2=\",round(theta2,1),\"degrees\"\n", "\n", "#NOTE:The answer in the textbook for principal stresses is off by 4 units in each case" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The principal stresses are as follows\n", "sigma1= 9606.0 psi and sigma2= 2394.0 psi\n", "The corresponding directions are\n", "Theta1= 28.2 degrees and Theta2= 118.2 degrees\n" ] } ], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.6, Page No:298" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "sigma_x=40 #Stress in x in MPa\n", "sigma_y=-100 #Stress in y in MPa\n", "tau_xy=-50 #Shear stress in MPa\n", "\n", "#Calculations\n", "tau_max=sqrt(((sigma_x-sigma_y)*0.5)**2+tau_xy**2) #Maximum in-plane shear in MPa\n", "\n", "#Orientation of Plane\n", "theta1=arctan(-((sigma_x-sigma_y)*(2*tau_xy)**-1))*180*pi**-1*0.5 #Angle in Degrees\n", "theta2=theta1+90 #Angle in degrees\n", "\n", "#Plane of max in-plane shear\n", "tau_x_y=-0.5*(sigma_x-sigma_y)*sin(2*theta1*pi*180**-1)+tau_xy*cos(2*theta1*pi*180**-1) \n", "\n", "#Normal Stress\n", "sigma=(sigma_x+sigma_y)*0.5 #Stress in MPa\n", "\n", "#Result\n", "print \"The maximum in-plane Shear is\",round(tau_x_y),\"MPa\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The maximum in-plane Shear is -86.0 MPa\n" ] } ], "prompt_number": 24 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.7, Page No:305" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Vairable Decleration\n", "sigma_x=40 #Stress in x in MPa\n", "sigma_y=20 #Stress in y in MPa\n", "tau_xy=16 #Shear in xy in MPa\n", "\n", "#Calculations\n", "sigma=(sigma_x+sigma_y)*0.5 #Normal Stress in MPa\n", "R=sqrt(((sigma_x-sigma_y)*0.5)**2+tau_xy**2) #Resultant Stress in MPa\n", "\n", "#Part 1\n", "sigma1=sigma+R #Principal Stress in MPa\n", "sigma2=sigma-R #Principal Stress in MPa\n", "theta=arctan(tau_xy*((sigma_x-sigma_y)*0.5)**-1)*180*pi**-1*0.5 #Orientation in degrees\n", "\n", "#Part 2\n", "tau_max=18.87 #From figure in MPa\n", "\n", "#Part 3\n", "sigma_xdash=sigma+tau_max*cos((100-theta*2)*pi*180**-1) #Stress in MPa\n", "sigma_ydash=sigma-tau_max*cos((100-theta*2)*pi*180**-1) #Stress in MPa\n", "tau_x_y=tau_max*sin((100-2*theta)*pi*180**-1) #Shear stress in MPa\n", "\n", "#Result\n", "print \"The principal Stresses are\"\n", "print \"Sigma1=\",round(sigma1,1),\"MPa and Sigma2=\",round(sigma2,1),\"MPa\"\n", "print \"The Principal Plane is at\",round(theta),\"degrees\"\n", "print \"The Maximum Shear Stress is\",tau_max,\"MPa\"\n", "print \"Sigma_x'=\",round(sigma_xdash),\"MPa and Sigma_y'=\",round(sigma_ydash,2),\"MPa\"\n", "print \"Tau_x'y'=\",round(tau_x_y,2),\"MPa\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The principal Stresses are\n", "Sigma1= 48.9 MPa and Sigma2= 11.1 MPa\n", "The Principal Plane is at 29.0 degrees\n", "The Maximum Shear Stress is 18.87 MPa\n", "Sigma_x'= 44.0 MPa and Sigma_y'= 15.98 MPa\n", "Tau_x'y'= 12.63 MPa\n" ] } ], "prompt_number": 32 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.9, Page No:316" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variabel Decleration\n", "sigma_w=120 #Working Stress in MPa\n", "tau_w=70 #Working Shear in MPa\n", "\n", "#Calcualtions\n", "#Section a-a\n", "M=3750 #Applied moment at section a-a in N.m\n", "T=1500 #Applied Torque at section a-a in N.m\n", "\n", "#After carrying out the variable based computation we compute d\n", "d1=((124.62)/(sigma_w*10**3*pi))**0.3333 #Diameter of the shaft in m\n", "d2=((65.6)/(tau_w*10**3*pi))**0.3333 #Diameter of the shaft in m\n", "d=max(d1,d2) #Diameter of the shaft to be selected in m\n", "\n", "#Result\n", "print \"The diameter of the shaft to be selected is\",round(d*1000,1),\"mm\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The diameter of the shaft to be selected is 69.2 mm\n" ] } ], "prompt_number": 37 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.10, Page No:318" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "t=0.01 #Thickness of the shaft in m\n", "p=2 #Internal Pressure in MPa\n", "r=0.45 #Mean radius of the vessel in m\n", "tw=50 #Working shear stress in MPa\n", "\n", "#Calculation\n", "sigma_x=(p*r)/(2*t) #Stress in MPa\n", "sigma_y=(p*r)/t #Stress in MPa\n", "\n", "R=100-67.5 #From the diagram in MPa\n", "tau_xy=sqrt((R**2-(sigma_y-67.5)**2)) #Stress in MPa\n", "\n", "J=2*pi*r**3*t #Polar Moment of inertia in mm^4\n", "\n", "T=1000*(tau_xy*J)/r #Maximum allowable Torque in kN.m\n", "\n", "#Result\n", "print \"The largest allowable Torque is\",round(T),\"kN.m\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The largest allowable Torque is 298.0 kN.m\n" ] } ], "prompt_number": 45 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.11, Page No:320" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "L=15 #Length of the shaft in inches\n", "r=3.0/8.001 #Radius of the shaft in inches\n", "T=540 #Torque applied in lb.in\n", "\n", "#Calculations\n", "V=30 #Transverse Shear Force in lb\n", "M=15*V #Bending Moment in lb.in\n", "I=(pi*r**4)/4.0 #Moment of Inertia in in^4\n", "J=2*I #Polar Moment Of Inertia in in^4\n", "\n", "#Part 1\n", "sigma=(M*r)/I #Bending Stress in psi\n", "tau_t=10**-3*(T*r)/J #Shear Stress in ksi\n", "\n", "sigma_max1=13.92 #From the Mohr Circle in ksi\n", "\n", "#Part 2\n", "Q=(2*r**3)/3.0 #First Moment in in^3\n", "b=2*r # in\n", "\n", "tau_V=10**-3*(V*Q)/(I*b) #Shear Stress in ksi\n", "tau=tau_t+tau_V #Total Shear in ksi\n", "\n", "sigma_max2=tau #Maximum stress in ksi\n", "\n", "#Result\n", "print \"The maximum normal stress in case 1 is\",sigma_max1,\"ksi\"\n", "print \"The Maximum normal stress in case 2 is\",round(sigma_max2,2),\"ksi\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The maximum normal stress in case 1 is 13.92 ksi\n", "The Maximum normal stress in case 2 is 6.61 ksi\n" ] } ], "prompt_number": 60 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.12, Page No:330" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "ex=800*10**-6 #Strain in x \n", "ey=200*10**-6 #Strain in y\n", "y_xy=-600*10**-6 #Strain in xy\n", "\n", "#Calculations\n", "e_bar=(ex+ey)*0.5 #Strain \n", "R=sqrt(2*300**2)*10**-6 #Resultant \n", "\n", "#Part 1\n", "e1=e_bar+R #Strain in Principal Axis\n", "e2=e_bar-R #Strain in Principal Axis\n", "\n", "#Part 2\n", "alpha=15*180**-1*pi #From the Mohr Circle in degrees\n", "e_xbar=e_bar-(R*cos(alpha)) #Strain in x \n", "e_ybar=e_bar+(R*cos(alpha)) #Strain in y\n", "\n", "shear_strain=2*R*sin(alpha) #Shear starin \n", "\n", "#Result\n", "print \"The principal Strains are\"\n", "print \"e1=\",round(e1,6),\"e2=\",round(e2,6)\n", "print \"The starin components are\"\n", "print \"ex'=\",round(e_xbar,6),\"ey'=\",round(e_ybar,6),\"y_x'y'=\",round(shear_strain,6)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The principal Strains are\n", "e1= 0.000924 e2= 7.6e-05\n", "The starin components are\n", "ex'= 9e-05 ey'= 0.00091 y_x'y'= 0.00022\n" ] } ], "prompt_number": 16 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.13, Page No:331" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "e_x=800*10**-6 #Strain in x\n", "e_y=200*10**-6 #Strain in y\n", "y_xy=-600*10**-6 #Strain in xy\n", "v=0.30 #Poissons Ratio\n", "E=200 #Youngs Modulus in GPa\n", "R_e=424.3*10**-6 #Strain\n", "e_bar=500*10**-6 #Strain\n", "\n", "#Calculations\n", "#Part 1\n", "R_sigma=10**-6*R_e*(E*10**9/(1+v)) #Stress in MPa\n", "sigma_bar=10**-6*e_bar*(E*10**9/(1-v)) #Stress in MPa\n", "\n", "#Part 2\n", "sigma1=sigma_bar+R_sigma #Principal Stress in MPa\n", "sigma2=sigma_bar-R_sigma #Principal Stress in MPa\n", "\n", "#Result\n", "print \"The principal Stresses are as follows\"\n", "print \"Sigma1=\",round(sigma1),\"MPa and Sigma2=\",round(sigma2,1),\"MPa\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The principal Stresses are as follows\n", "Sigma1= 208.0 MPa and Sigma2= 77.6 MPa\n" ] } ], "prompt_number": 20 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 8.8.14, Page No:336" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "#Variable Decleration\n", "e_a=100*10**-6 #Strain \n", "e_b=300*10**-6 #Strain\n", "e_c=-200*10**-6 #Strain\n", "E=180 #Youngs Modulus in GPa\n", "v=0.28 #Poissons Ratio \n", "\n", "#Calculations\n", "y_xy=(e_b-(e_a+e_c)*0.5) #Strain in xy\n", "e_bar=(e_a+e_c)*0.5 #Strain \n", "R_e=sqrt(y_xy**2+(150*10**-6)**2) #Resultant Strain\n", "\n", "#Corresponding Parameters from Mohrs Diagram\n", "sigma_bar=(E/(1-v))*e_bar*10**3 #Stress in MPa\n", "R_sigma=(E/(1+v))*R_e*10**3 #Resultant Stress in MPa\n", "\n", "#Principal Stresses\n", "sigma1=sigma_bar+R_sigma #MPa\n", "sigma2=sigma_bar-R_sigma #MPa\n", "\n", "theta=arctan(y_xy/(150*10**-6))*180*pi**-1*0.5 #Degrees\n", "\n", "#Result\n", "print \"The Principal Stresses are as follows\"\n", "print \"Sigma1=\",round(sigma1,1),\"MPa and Sigma2=\",round(sigma2,2),\"MPa\"\n", "print \"The plane orientation is\",round(theta,2),\"degrees\"" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The Principal Stresses are as follows\n", "Sigma1= 41.0 MPa and Sigma2= -66.05 MPa\n", "The plane orientation is 33.4 degrees\n" ] } ], "prompt_number": 32 } ], "metadata": {} } ] }