{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "# Chapter 2 : Stress and Strain—Axial Loading" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.01, Page number 69" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Deformation of the steel rod = 0.000000 in\n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "E=29*(pow(10,6)) # Modulus of elasticity(psi) \n", "L1=12 # Length(in) \n", "L2=12 # Length(in)\n", "L3=16 # Length(in)\n", "A1=0.9 # Area(in**2) \n", "A2=0.9 # Area(in**2)\n", "A3=0.3 # Area(in**2) \n", "P1=60*(pow(10,3)) # Internal force(lb)\n", "P2=15*(pow(10,3)) # Internal force(lb)\n", "P3=30*(pow(10,3)) # Internal force(lb)\n", "\n", "#Calculation \n", "Delta=round((1/E)*(((P1*L1)/A1)+(-(P2*L2)/A2)+((P3*L3)/A3)),4) # deformation of the steel rod(in)\n", "\n", "#Result\n", "print ('Deformation of the steel rod = %lf in' %Delta)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 2.1, Page number 70" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a): Deflection of B = -0.514286 psi\n", "Case(b): Deflection of D = 0.300000 psi\n", "Case(c): Deflection of E = 1.928223 psi\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "Eab=70*(pow(10,9)) # Modulus of elasticity of AB(GPa) \n", "Aab=500*(pow(10,-6)) # Cross sectional area of AB(mm**2) \n", "Ecd=200*(pow(10,9)) # Modulus of elasticity of CD(GPa)\n", "Acd=600*(pow(10,-6)) # Cross sectional area of CD(mm**2) \n", "Fe=30 # Force(kN) \n", "Pb=-60*(pow(10,3)) # Internal force in link AB(kN)\n", "n=-1\n", "Pd=90*(pow(10,3)) # Internal force in link CD(kN)\n", "Lab=0.3 # Length of AB(m)\n", "Lcd=0.4 # Length of CD(m)\n", "x=symbols('x') # Variable declaration \n", "Se=symbols('Se') # Variable declaration\n", "\n", "#Calculation \n", "#Free Body: Bar BDE\n", "Fcd=(30*0.6)/(0.2) # Force of tension(kN)\n", "Fab=n*(30*0.4)/(0.2) # Force of compression(kN)\n", "\n", "#Case(a)\n", "#Deflection of B.\n", "Sb=((Pb*Lab)/(Aab*Eab))*(1000) # Deflection of B(mm)\n", "\n", "#Case(b)\n", "#Deflection of D.\n", "Sd=((Pd*Lcd)/(Acd*Ecd))*(1000) # Deflection of D(mm)\n", "\n", "#Case(c)\n", "#Deflection of E.\n", "x=solve(((200-x)/x)-((0.514)/(0.300)),x) # Distance HD(mm)\n", "Se=solve(((400+73.7)/73.7)-(Se/0.300),Se) # Deflection of E(mm)\n", "\n", "#Result\n", "print('Case(a): Deflection of B = %1f psi' %Sb)\n", "print('Case(b): Deflection of D = %lf psi' %Sd)\n", "print('Case(c): Deflection of E = %lf psi' %Se[0])" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## SAMPLE PROBLEM 2.2, Page number 71" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Stress in rod = 10.533169 ksi\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "Lb=18 # Length of Bolt CD(in)\n", "Pb=symbols('Pb') # Force(kips)\n", "Ab=((1/4.0)*(math.pi)*pow(0.75,2)) # Area of cross section of CD and GH(in**2) \n", "Eb=(29*(pow(10,6))) # Modulus of elasticity of steel(psi)\n", "Lr=12 # Length of Bolt EF(in)\n", "Pr=symbols('Pr') # Force(kips)\n", "Ar=((1/4.0)*(math.pi)*pow(1.5,2)) # Area of cross section of EF(in**2) \n", "Er=((10.6)*(pow(10,6))) # Modulus of elasticity of aluminium(psi)\n", "n=-1\n", "\n", "#Calculation \n", "# Bolts CD and GH\n", "Sb=((Pb*Lb)/(Ab*Eb)) # Deformation \n", "\n", "# Rod EF\n", "Sr=n*((Pr*Lr)/(Ar*Er)) # Deformation\n", "\n", "# Displacement of D Relative to B\n", "Eq=(1/4.0)*(0.1)-Sb+Sr # Because displacements are equal to the deformations of the bolts and of the rod\n", "\n", "# Free Body: Casting B\n", "#Pr-2*Pb=0 # Submission of forces is zero\n", "\n", "# Forces in Bolts and Rod\n", "Pb=(solve((Eq).subs(Pr,(2*Pb)))[0])/(1000.0) # Force in the bolt(kips) \n", "Pr=2*Pb # Force in th rod(kips) \n", "\n", "# Stress in Rod\n", "er=Pr/Ar # Stress in rod(ksi)\n", "\n", "\n", "#Result\n", "print('Stress in rod = %1f ksi' %er)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.02, Page number 78" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Deformation in the rod :-\n", "L*P/(A1*E1 + A2*E2)\n", "Deformation in the tube :-\n", "L*P/(A1*E1 + A2*E2)\n" ] } ], "source": [ "import math\n", "from sympy import symbols, solve\n", "\n", "#Variable declaration\n", "P1=symbols('P1') # Axial force \n", "L=symbols('L') # Length of rod\n", "A1=symbols('A1') # Cross section of rod\n", "E1=symbols('E1') # Modulus of elasticity\n", "P2=symbols('P2') # Axial force\n", "A2=symbols('A2') # Cross section of rod\n", "E2=symbols('E2') # Modulus of elasticity \n", "P=symbols('P') # Total axial force \n", "\n", "\n", "#Calculation \n", "S1=(P1*L)/(A1*E1) # Deforamation in rod \n", "S2=(P2*L)/(A2*E2) # Deformation in rod\n", "\n", "P1=((P2)/(A2*E2))*(A1*E1) # Equating deformations S1 and S2 \n", "\n", "P2=solve(P1-P+P2,P2) # Solving the above equation for P2\n", "P1=((P2[0]/(A2*E2))*(A1*E1)) # Solving for P1 \n", "S1=(P1*L)/(A1*E1)\n", "S2=(P2[0]*L)/(A2*E2)\n", "\n", "#Result\n", "print(\"Deformation in the rod :-\")\n", "print(S1)\n", "print(\"Deformation in the tube :-\")\n", "print(S2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.03, Page number 79" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Stress in AC :-\n", "L1*P1/(A*E)\n", "Stress in BC :-\n", "L2*P2/(A*E)\n" ] } ], "source": [ "import math\n", "from sympy import symbols,solve\n", "\n", "#Variable declaration\n", "Ra=symbols('Ra') # Reaction at A\n", "Rb=symbols('Rb') # Reaction at B\n", "P=symbols('P') # Net reaction \n", "S=symbols('S') # Total elongation\n", "S1=symbols('S1') # Elongation of AC\n", "S2=symbols('S2') # Elongation of BC\n", "L1=symbols('L1') # Length of AC\n", "L2=symbols('L2') # Length of BC\n", "P1=symbols('P1') # Internal force \n", "P2=symbols('P2') # Internal force\n", "A=symbols('A') # Area of cross section\n", "E=symbols('E') # Modulus of elasticity \n", "\n", "\n", "#Calculation \n", "S1=(P1*L1)/(A*E) # Elongation of AC \n", "S2=(P2*L2)/(A*E) # Elongation of BC\n", "S=S1+S2 # Total elongation of the bar \n", "S=0 # Total elongation of the bar is zero \n", "\n", "x=solve((Ra*L1-Rb*L2, Ra+Rb-P), Ra, Rb) # Solving the equations to find reactions at A and B. \n", "\n", "e1=(x[Ra])/A # Stress in AC\n", "e2=(x[Rb])/A # Stress in BC\n", "\n", "#Result\n", "print(\"Stress in AC :-\")\n", "print(S1)\n", "print(\"Stress in BC :-\")\n", "print(S2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.04, Page number 80" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Reaction at A = 577823.076923\n", "Reaction at B = 576923.076923\n" ] } ], "source": [ "import math\n", "from sympy import symbols, solve\n", "\n", "#Variable declaration\n", "P1=0 # Force in the first portion(N) \n", "P2=600*(pow(10,3)) # Force in the second portion(N)\n", "P3=600*(pow(10,3)) # Force in the third portion(N)\n", "P4=900*(pow(10,3)) # Force in the forth portion(N) \n", "E=symbols('E') # Modulus of elasticity\n", "Rb=symbols('Rb') # Reaction at B(N) \n", "n=-1\n", "\n", "A1=400*(pow(10,-6)) # Area of cross section of first portion(m**2)\n", "A2=400*(pow(10,-6)) # Area of cross section of second portion(m**2) \n", "A3=250*(pow(10,-6)) # Area of cross section of third portion(m**2)\n", "A4=250*(pow(10,-6)) # Area of cross section of forth portion(m**2)\n", "\n", "L1=0.150 # Length of first portion(m) \n", "L2=0.150 # Length of second portion(m)\n", "L3=0.150 # Length of third portion(m)\n", "L4=0.150 # Length of forth portion(m)\n", "\n", "#Calculation \n", "Sl=((600*(pow(10,3)))/(400*(pow(10,-6))) + (600*(pow(10,3)))/(250*(pow(10,-6))) + (900*(pow(10,3)))/(250*(pow(10,-6))))*(0.150/E) # Deformation(m) \n", "P1=n*Rb # Force on first portion(N) \n", "P2=n*Rb # Force on second portion(N) \n", "A1=400*(pow(10,-6)) # Area of cross section of first portion(m**2)\n", "A2=250*(pow(10,-6)) # Area of cross section of second portion(m**2)\n", "L1=0.300 # Length of first portion(m) \n", "L2=0.300 # Length of second portion(m)\n", "\n", "Sr=(P1*L1)/(A1*E) + (P2*L2)/(A2*E) # Deformation(m)\n", "\n", "Rb=solve(Sl+Sr,Rb) # Reaction at B(N)\n", "\n", "Ra=900+(Rb[0]) # Reaction at A(N) \n", "\n", "#Result\n", "print('Reaction at A = %lf' %Ra)\n", "print('Reaction at B = %lf' %Rb[0])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.05, Page number 81" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Reaction at A = 784.615385\n", "Reaction at B = 115.384615\n" ] } ], "source": [ "import math\n", "from sympy import symbols, solve\n", "\n", "#Variable declaration\n", "E=200*(pow(10,9)) # Modulus of elasticity(GPa)\n", "Rb=symbols('Rb') # Reaction at B(N) \n", "Ra=symbols('Ra') # Reaction at A(N)\n", "#Calculation \n", "Rb=(solve(((1.125*(pow(10,9)))/(200*(pow(10,9))))-(((1.95*(pow(10,3)))*Rb)/(200*(pow(10,9))))-(4.5*(pow(10,-3))),Rb))[0]/(10**3) # Solving for Rb(N)\n", "Ra=solve(Ra-900+Rb,Ra) # Solving for Ra(N) \n", "\n", "#Result\n", "print('Reaction at A = %lf' %Ra[0])\n", "print('Reaction at B = %lf' %Rb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.06, Page number 84" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Stress in portion AC = 31.416667\n", "Stress in portion CB = 15.708333\n" ] } ], "source": [ "import math\n", "from sympy import symbols, solve\n", "\n", "#Variable declaration\n", "E=29*(pow(10,6))*(1.0) # Modulus of elasticity(psi)\n", "Alpha=6.5*(pow(10,-6)) # Constant(/F)\n", "n=-1\n", "L1=12 # Length of AC(m)\n", "L2=12 # Length of CB(m)\n", "A1=0.6 # Area of cross section of AC(m**2)\n", "A2=1.2 # Area of cross section of BC(m**2) \n", "Rb=symbols('Rb') # Reaction at B(N)\n", "P1=Rb # Internal force(N)\n", "P2=Rb # Internal force(N)\n", "DELTAt=n*(50+75) # Diff in temperature(F)\n", "AC=12 # Length of AC(m)\n", "CB=12 # Length of CB(m)\n", "\n", "#Calculation\n", "St=Alpha*DELTAt*24 # Deformation(in)\n", "Sr=((P1*L1)/(A1*E)) + ((P2*L2)/(A2*E)) # Deformation(in)\n", "\n", "Rb=(solve(St+Sr,Rb))[0]/(1000.0) # Reaction at B(kips) \n", "S1=(Rb)/A1 # Stress in portion AC(ksi)\n", "S2=(Rb)/A2 # Stress in portion CB(ksi)\n", "\n", "Et=Alpha*DELTAt # Thermal strain(in./in.) \n", "Ratio=S1/E # Other component of eAC(in./in.)\n", "Eac=Et + S1/E # Component of strain in AC(in)\n", "Ecb=Et + S2/E # Component of strain in CB(in) \n", "\n", "Sac=Eac*(AC) # Deformation of rod AC(in)\n", "Scb=Ecb*(CB) # Deformation of rod CB(in)\n", "\n", "#Result\n", "print('Stress in portion AC = %lf' %S1)\n", "print('Stress in portion CB = %lf' %S2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 2.3, Page number 86" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case(a): Force in rod DF = 7.5 psi\n", "Case(a): Force in rod CE = 2.497916 psi\n", "Case(b): Deflection of point A = 0.043249 psi\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "Ldf=30 # Length of DF(in)\n", "Adf=(1/4.0)*(math.pi)*(3/4.0)*(3/4.0) # Cross sectional area of DF(in**2) \n", "Lce=24 # Length of CE(in)\n", "Ace=(1/4.0)*(math.pi)*(1/2.0)*(1/2.0) # Cross sectional area of DF(in**2) \n", "E=(10.6*pow(10,6)) # Molulus of elasticity(psi)\n", "Fce=symbols('Fce') # Force on CE(N) \n", "Fdf=symbols('Fdf') # Force on DF(N)\n", "Sd=symbols('Sd') # Force on CE(N) \n", "\n", "\n", "#Calculation \n", "# Case(a)\n", "# Statics\n", "Eq=Fce*12 + Fdf*20 - 180 # Because submission of Mb is 0 \n", "\n", "# Geometry\n", "Sc=0.6*Sd # By using similar triangles\n", "Sa=0.9*Sd # By using similar triangles\n", "\n", "# Deformations\n", "Sc=(Fce*Lce)/(Ace*E) # Deformation at C(in) \n", "Sd=(Fdf*Ldf)/(Adf*E) # Deformation at D(in) \n", "Fce=0.333*Fdf # By solving Sc= 0.6*Sd we get this result\n", "\n", "# Force in Each Rod\n", "Fdf=(180/(((12)*(0.333))+20)) # By using Eq and the value of Fce \n", "Fce=0.333*Fdf # Force in CE(N)\n", "\n", "# Case(b)\n", "# Deflections\n", "Sd=(Fdf*(pow(10,3))*Ldf)/(Adf*E) # Deflection of point D(in)\n", "Sa=0.9*Sd # Deflection of point A(in) \n", "\n", "\n", "#Result\n", "print('Case(a): Force in rod DF = %.1f psi' %Fdf)\n", "print('Case(a): Force in rod CE = %lf psi' %Fce)\n", "print('Case(b): Deflection of point A = %lf psi' %Sa)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 2.4, Page number 87" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Stress in cylinder = 44.8 MPa\n" ] } ], "source": [ "import math\n", "from sympy import symbols\n", "\n", "#Variable declaration\n", "Ldf=30 # Length of DF(in)\n", "Adf=(1/4.0)*(math.pi)*(3/4.0)*(3/4.0) # Cross sectional area of DF(in**2) \n", "Lce=24 # Length of CE(in)\n", "Ace=(1/4.0)*(math.pi)*(1/2.0)*(1/2.0) # Cross sectional area of DF(in**2) \n", "E=(10.6*pow(10,6)) # Molulus of elasticity(psi)\n", "Fce=symbols('Fce') # Force on CE(N) \n", "Fdf=symbols('Fdf') # Force on DF(N)\n", "Sd=symbols('Sd') # Force on CE(N) \n", "Rb=symbols('Rb') # Reaction at B(m) \n", "\n", "\n", "#Calculation \n", "Ra=0.4*Rb # By considering the free body motion\n", "L=0.3 # Length of rod DE(m) \n", "DeltaT=30 # Change in temperature(C) \n", "Alpha=(20.9*pow(10,-6)) # Constant(/C)\n", "A=(1/4)*(math.pi)*(pow(0.22,2)) # Area of cross-section of rod AC(m**2)\n", "E=200 # Molulus of elasticity(psi)\n", "\n", "St=L*(DeltaT)*Alpha # Deflection(m)\n", "\n", "temp=(L/(A*E)) \n", "Sc=str(temp)+'*Ra' # Deflection(m) \n", "\n", "Sd=str(0.4)+'*Sc' # Deflection(m)\n", "Sd=4.74*pow(10,-9) # After computing the above equation\n", "\n", "Sbd=str(4.04*pow(10,-9))+'*Ra' # Deflection(m)\n", "\n", "S1=str(5.94*pow(10,-9))+'*Rb' # Deflection(m)\n", "\n", "Rb=((188.1*pow(10,-6))/(5.94*pow(10,-9)))/(pow(10,3)) # Reaction at B(m) \n", "\n", "db=((round(Rb,1))/((1/4.0)*(math.pi)*(pow(0.03,2))))/(pow(10,3)) # Stress in cylinder(MPa)\n", "\n", "#Result\n", "print('Stress in cylinder = %.1f MPa' %db)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.07, Page number 94" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Modulus of elasticity = 99471839.4 GPa\n", "Poissons ratio = 0.2 \n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "P=12*(pow(10,3)) # Axial load(kN)\n", "r=8*(pow(10,-3)) # Radius of the rod(m)\n", "n=-1\n", "\n", "#Calculation\n", "A=(math.pi)*(r**2) # Cross sectional area of rod(m**2)\n", "Sx=(P/A) # Stress in cylinder(MPa) \n", "Ex=(300/500.0) # Strain()\n", "Ey=(n*(2.4))/16.0 # Strain()\n", "\n", "E=Sx/Ex # Modulus of elasticity(GPa)\n", "v=n*(Ey/Ex) # Poissons ratio() \n", "\n", "#Result\n", "print('Modulus of elasticity = %.1f GPa' %E)\n", "print('Poissons ratio = %.1f ' %v)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.08, Page number 96" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change in length in y direction = -0.000600 in\n", "Change in length in z direction = -0.000900 in\n", "Pressure = 20.7 ksi\n" ] } ], "source": [ "import math\n", "from sympy import symbols, solve\n", "\n", "#Variable declaration\n", "E=29*(pow(10,6)) # Modulus of elasticity(GPa)\n", "v=0.29 # Poissons ratio()\n", "Sx=symbols('Sx') # Strain variable \n", "Sy=symbols('Sy') # Strain variable\n", "Sz=symbols('Sz') # Strain variable\n", "Ex=symbols('Ex') # Stress variable\n", "Ey=symbols('Ey') # Stress variable\n", "Ez=symbols('Ez') # Stress variable\n", "p=symbols('p') # pressure\n", "n=-1\n", "Sx=n*p # Strain at x\n", "Sy=n*p # Strain at y\n", "Sz=n*p # Strain at z \n", "Ey=Ez=Ex # Equating all the stresses\n", "\n", "#Calculation\n", "# Case(a)\n", "Ex=((n*p)*(1-(2*v)))/E # Stress \n", "Ex=(n*1.2*(pow(10,3)))/4 # Stress(in./in)\n", "DELTAy=n*(300*(pow(10,-6)))*(2) # Change in length(in)\n", "DELTAz=n*(300*(pow(10,-6)))*(3) # Change in length(in) \n", "# Case(b)\n", "p=(solve(p+((29*(pow(10,6)))*(n*300*(pow(10,-6))))/(1-0.58),p)[0])/(pow(10,3)) # pressure(ksi)\n", "\n", "#Result\n", "print('Change in length in y direction = %1f in' %DELTAy)\n", "print('Change in length in z direction = %1f in' %DELTAz)\n", "print('Pressure = %.1f ksi' %p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.09, Page number 98" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change in volume = -217.728000 mm**3\n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "p=180 # Hydrostatic pressure(MPa)\n", "E=200 # Modulus of elasticity(GPa)\n", "v=0.29 # Poissons ratio() \n", "\n", "#Calculation\n", "k=E/(3*(1-(2*v))) # Bulk modulus of steel(GPa)\n", "e=-p/k # Dialation \n", "V=80*40*60 # Volume of block in unstressed state(mm**3)\n", "DELTAv=(e*V)/(pow(10,3)) # change in volume per unit volume\n", "\n", "# Results\n", "print('Change in volume = %1f mm**3' %DELTAv)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.10, Page number 101" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Shearing strain in rod=0.020000 rad\n", "Force exerted on the upper plate=36.000000 kips\n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "G=90 # Modulus of rigidity(ksi)\n", "disp=0.04 # Displacement of upper rod(in)\n", "Lda=2 # Height of bar(in)\n", "A=8*2.5 # Area of cross section(in**2)\n", "\n", "#Calculation\n", "Yxy=(disp/Lda) # Shearing strain(rad)\n", "Txy=(90*(pow(10,3)))*(0.020) # Shearing stress(psi) \n", "P=(Txy*A)/(pow(10,3)) # Force exerted on the upper plate(kips) \n", "\n", "# Results\n", "print('Shearing strain in rod=%1f rad' %Yxy)\n", "print('Force exerted on the upper plate=%1f kips' %P)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.11, Page number 105" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change in cube dimension in x direction=-14.981914 um\n", "Change in cube dimension in y direction=0.000000 um\n", "Change in cube dimension in z direction=5.443356 um\n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "Ex=155.0 # Modulus of elasticity in x direction(GPa)\n", "Ey=12.10 # Modulus of elasticity in y direction(GPa)\n", "Ez=12.10 # Modulus of elasticity in z direction(GPa)\n", "Vxy=0.248 # Poissons ratio in xy direction\n", "Vxz=0.248 # Poissons ratio in xz direction\n", "Vyz=0.458 # Poissons ratio in yz direction\n", "n=-1\n", "F=140*(pow(10,3)) # Compressive load(kN)\n", "L=0.060 # Length of cube(m)\n", "\n", "#Calculation\n", "#(a) Free in y and z Directions\n", "Sx=(n*F)/(0.060*0.060) # Stress in x direction(MPa)\n", "Sy=0 # Stress in y direction(MPa) \n", "Sz=0 # Stress in z direction(MPa) \n", "ex=Sx/Ex # Lateral strains \n", "ey=n*((Vxy*Sx)/Ex) # Lateral strains \n", "ez=n*((Vxy*Sx)/Ex) # Lateral strains\n", "DELTAx=ex*L # Change in cube dimension in x direction(um) \n", "DELTAy=ey*L # Change in cube dimension in y direction(um)\n", "DELTAz=ez*L # Change in cube dimension in z direction(um)\n", "#(b) Free in z Direction, Restrained in y Direction\n", "Sx=n*38.89 # Stress in x direction(MPa)\n", "Sy=(Ey/Ex)*(Vxy)*(Sx) # Stress in y direction(MPa) \n", "Vyx=(Ey/Ex)*(Vxy) # Poissons ratio\n", "ex=(Sx/Ex)-(((Vyx)*(Sy))/Ey) # Lateral strains in x direction\n", "ey=0 # Lateral strains in y direction\n", "ez=n*((Vxz*Sx)/Ex)-(((Vyz)*(Sy))/Ey) # Lateral strains in z direction\n", "DELTAx=ex*L*1000 # Change in cube dimension in x direction(um) \n", "DELTAy=ey*L # Change in cube dimension in y direction(um)\n", "DELTAz=ez*L*1000 # Change in cube dimension in z direction(um)\n", "\n", "# Results\n", "print('Change in cube dimension in x direction=%1f um' %DELTAx)\n", "print('Change in cube dimension in y direction=%1f um' %DELTAy)\n", "print('Change in cube dimension in z direction=%1f um' %DELTAz)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 2.5, Page number 107" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Change in diamter of rod AB =0.004800 in\n", "Change in diamter of rod CD =0.014400 in\n", "Change in thickness =-0.000800 in\n", "Volume of the plate =0.180000 in**3\n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "d=9 # Diameter of the rod(in)\n", "t=3/4.0 # Thickness of the rod(in)\n", "ex=12 # Normal stresses(ksi) \n", "ez=20 # Normal stresses(ksi)\n", "E=(10*pow(10,6)) # Moduluus of elasticity(psi)\n", "v=(1/3) # Poissons ratio\n", "V=15*15*(3/4.0) # Volume(in**3)\n", "n=-1\n", "\n", "#Calculation\n", "STRAINx=(1/(pow(10,7)*(1.0)))*(12-(20/3.0))*(1000) # Strain in x direction(in./in) \n", "STRAINy=n*(1/(pow(10,7)*1.0))*((12/3.0)+(20/3.0))*(1000) # Strain in y direction(in./in)\n", "STRAINz=(1/(pow(10,7)*(1.0)))*(20-(12/3.0))*(1000) # Strain in z direction(in./in)\n", "\n", "\n", "#Case(a) \n", "DELTAba=(STRAINx)*(d) # Change in diameter(in)\n", "#Case(b) \n", "DELTAcd=(STRAINz)*(d) # Change in diameter(in)\n", "#Case(c) \n", "DELTAt=(STRAINy)*(t) # Change in thickness(in)\n", "#Case(d) \n", "e=(STRAINx+STRAINy+STRAINz) # Volume of the plate(in**3)\n", "DeltaV=(e*V) \n", "\n", "# Results\n", "print('Change in diamter of rod AB =%1f in' %DELTAba)\n", "print('Change in diamter of rod CD =%1f in' %DELTAcd)\n", "print('Change in thickness =%1f in' %DELTAt)\n", "print('Volume of the plate =%1f in**3' %DeltaV)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.12, Page number 116" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Largest Axial Load =36.300000 in\n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "D=60 # Width(mm) \n", "d=40 # Width(mm)\n", "r=8 # Radius(mm)\n", "K=1.82 # Stress-concentration factor\n", "Smax=165 # Allowable normal stress(MPa)\n", "\n", "#Calculation\n", "eave=(165/1.82) # Average stress in the narrower portion(MPa) \n", "P=round((40*10*eave)/(1000),1) # Largest Axial Load(kN)\n", "\n", "# Results\n", "print('Largest Axial Load =%1f in' %P)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2.13, Page number 117" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Permanent set deformation =6.250000 mm\n" ] } ], "source": [ "import math\n", "\n", "#Variable declaration\n", "L=500.0 # Length of rod(mm)\n", "A=60 # Cross Sectional area(mm**2)\n", "E=200 # Modulus of elasticity(GPa)\n", "ey=300 # Yield Point(MPa) \n", "DELTAc=7 # Stretch(mm) \n", "\n", "#Calculation\n", "ec=DELTAc/L # Maximum strain permitted on point C\n", "ey=(ey*(pow(10,6)))/(E*(pow(10,9))*(1.0)) # Maximum strain permitted on point Y \n", "ed=ec-ey # Strain after unloading\n", "DELTAd=ed*L # Deformation(mm)\n", "\n", "# Results\n", "print('Permanent set deformation =%1f mm' %DELTAd) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SAMPLE PROBLEM 2.6, Page number 123" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Required maximum value of Q =240.0 mm\n", "The final position of the beam =11.000000 mm\n" ] } ], "source": [ "from sympy import symbols, solve\n", "\n", "#Variable declaration\n", "Lad=2 # Length of AD(m)\n", "Lce=5 # Length of CE(m) \n", "E=200.0 # Modulus of elasticity(GPa)\n", "ey=300 # Stress(MPa)\n", "Aad=400 # Area of cross section of AD(mm**2)\n", "Ace=500.0 # Area of cross section of CE(mm**2)\n", "STRAINad=ey/E # Stress(MPa) # \n", "\n", "#Calculation\n", "Pad=symbols('Pad') # Variable declaration\n", "Pce=symbols('Pce') # Variable declaration\n", "Pce=Pad\n", "Q=symbols('Q') # Variable declaration\n", "Q=2*Pad # Force(kN) \n", "d=ey # Maximum elastic deflection of point A\n", "PadM=ey*Aad # Maximum force(kN) \n", "Q=(2*PadM)/(1000.0) # Maximum force(kN)\n", "Sa1=STRAINad*Lad # Maximum deflection of point A(mm)\n", "Pce=120 # This implies Pad is also 120\n", "ece=Pce/Ace # Stress in rod CE(MPa)\n", "STRAINce=ece/E # Strain CE\n", "Sc1=STRAINce*Lce # Deflection of point C(mm)\n", "Sb=(1/2)*(Sa1+Sc1) # Deflection of point B(mm)\n", "\n", "# Since we must have Sb 10 mm, we conclude that plastic deformation will occur.\n", "\n", "# Plastic Deformation\n", "eAD=ey # Equating stresses\n", "Sc=6 # Deflection in C(mm)\n", "Sa2=symbols('Sa2') # Variable declaration \n", "Sa2=solve(((1/2)*(Sa2+6)-10),Sa2) # The deflection Sa for which Sb = 10\n", "\n", "# Unloading\n", "Sa3=14-3 #Since the stress in rod CE remained within the elastic range, we note that the final deflection of point C is zero.\n", "\n", "print('Required maximum value of Q =%.1f mm' %Q)\n", "print('The final position of the beam =%1f mm' %Sa3)" ] } ], "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 }